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Geometry and Trigonometry
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Question 1 1 of 269 selected Lines, Angles, And Triangles H

  • Triangle upper A upper C upper D partially overlaps triangle upper E upper B upper D.
  • Point upper C is on line segment upper B upper D.
  • Point upper E is on line segment upper A upper D.
  • The measure of angle upper A upper E upper B is x°.
  • A note indicates the figure is not drawn to scale.

 

In the figure, A C = C D . The measure of angle EBC is 45°, and the measure of angle ACD is 104°. What is the value of x ?

Show Answer Correct Answer: 83

The correct answer is 83 . It's given that in the figure, A C = C D . Thus, triangle A C D is an isosceles triangle and the measure of angle CDA is equal to the measure of angle CAD. The sum of the measures of the interior angles of a triangle is 180°. Thus, the sum of the measures of the interior angles of triangle A C D is 180°. It's given that the measure of angle A C D is 104°. It follows that the sum of the measures of angles CDA and CAD is (180-104)°, or 76°. Since the measure of angle CDA is equal to the measure of angle CAD, the measure of angle CDA is half of 76°, or 38°. The sum of the measures of the interior angles of triangle B D E is 180°. It's given that the measure of angle EBC is 45°. Since the measure of angle BDE, which is the same angle as angle CDA, is 38°, it follows that the measure of angle DEB is (180-45-38)°, or 97°. Since angle DEB and angle AEB form a straight line, the sum of the measures of these angles is 180°. It's given in the figure that the measure of angle AEB is x°. It follows that 97+x=180. Subtracting 97 from both sides of this equation yields x = 83 .

Question 2 2 of 269 selected Lines, Angles, And Triangles E

Triangles E F G and J K L are congruent, where E , F , and G correspond to J , K , and L , respectively. The measure of angle E is 45° and the measure of angle F is 20°. What is the measure of angle J ?

  1. 20°

  2. 45°

  3. 135°

  4. 160°

Show Answer Correct Answer: B

Choice B is correct. It's given that triangles E F G and J K L are congruent such that angle E corresponds to angle J . Corresponding angles of congruent triangles are congruent, so angle E and angle J must be congruent. Therefore, if the measure of angle E is 45°, then the measure of angle J is also 45°.

Choice A is incorrect. This is the measure of angle K , not angle J .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 3 3 of 269 selected Area And Volume M

The length of each edge of a box is 29 inches. Each side of the box is in the shape of a square. The box does not have a lid. What is the exterior surface area, in square inches, of this box without a lid?

Show Answer Correct Answer: 4205

The correct answer is 4,205 . The exterior surface area of a figure is the sum of the areas of all its faces. It's given that the box does not have a lid and that each side of the box is in the shape of a square. Therefore, the box consists of 5 congruent square faces. It's also given that the length of each edge is 29 inches. Let s represent the length of an edge of a square. It follows that the area of a square is equal to s 2 . Therefore, the area of each of the 5 square faces is equal to 292, or 841 , square inches. Since the box consists of 5 congruent square faces, it follows that the sum of the areas of all its faces, or the exterior surface area of this box without a lid, is 5(841), or 4,205 , square inches.

Question 4 4 of 269 selected Lines, Angles, And Triangles H

  • From left to right, horizontal line segment upper P upper V includes the following points:
    • Upper P
    • Upper Q
    • Upper R
    • Upper S
    • Upper T
    • Upper V
  • Points upper X and upper U are each below line segment upper P upper V.
  • From left to right, the following 2 overlapping triangles are formed between points on line segment upper P upper V and points upper X and upper U:
    • Triangle upper Q upper X upper S
    • Triangle upper R upper U upper T
  • Line segments upper R upper U and upper S upper X intersect at point upper W.
  • A note indicates the figure is not drawn to scale.

In the figure shown, points Q , R , S , and T lie on line segment P V , and line segment R U intersects line segment S X at point W . The measure of SQX is 48°, the measure of SXQ is 86°, the measure of SWU is 85°, and the measure of VTU is 162°. What is the measure, in degrees, of TUR

Show Answer Correct Answer: 123

The correct answer is 123 . The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is 180  degrees. It's given that the measure of SQX is 48° and the measure of SXQ is 86°. Since points S , Q , and X form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of QSX is 180-48-86, or 46 . It's also given that the measure of SWU is 85°. Since SWU and SWR are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of SWR is 180-85, or 95 . Since points R , S , and W form a triangle, and RSW is the same angle as QSX, it follows from the triangle angle sum theorem that the measure, in degrees, of WRS is 180-46-95, or 39 . It's given that the measure of VTU is 162°. Since VTU and STU are supplementary angles, the sum of their measures is 180 degrees. It follows that the measure, in degrees, of STU is 180-162, or 18. Since points R , T , and U form a triangle, and URT is the same angle as WRS, it follows from the triangle angle sum theorem that the measure, in degrees, of TUR is 180-39-18, or 123.

Question 5 5 of 269 selected Right Triangles And Trigonometry M

Triangle F G H is similar to triangle J K L , where angle F corresponds to angle J and angles G and K are right angles. If sin(F)=308317, what is the value of sin(J)?

  1. 75 317

  2. 308 317

  3. 317 308

  4. 317 75

Show Answer Correct Answer: B

Choice B is correct. If two triangles are similar, then their corresponding angles are congruent. It's given that right triangle F G H is similar to right triangle J K L and angle F corresponds to angle J . It follows that angle F is congruent to angle J and, therefore, the measure of angle F is equal to the measure of angle J . The sine ratios of angles of equal measure are equal. Since the measure of angle F is equal to the measure of angle J , sin(F)=sin(J). It's given that sin(F)=308317. Therefore,  sin(J) is 308317.

Choice A is incorrect. This is the value of cos(J), not the value of sin(J).

Choice C is incorrect. This is the reciprocal of the value of sin(J), not the value of sin(J).

Choice D is incorrect. This is the reciprocal of the value of cos(J), not the value of sin(J).

Question 6 6 of 269 selected Lines, Angles, And Triangles H

Intersecting lines r, s, and t are shown below.

The figure presents three intersecting lines, labeled r, s, and t. Line s begins near the top of the figure and extends downward and slightly to the left. Line t begins to the left of line s, and extends downward and to the right, intersecting line s. Line r begins below line t and to the left of line s. Line r extends upward and to the right, first intersecting line s and then line t. The following angle measures are indicated: the angle formed to the right of line s and above line t measures 106 degrees, and the angle formed to the left of line s and above line r measures x degrees. The angle formed below line t and above line r measures 23 degrees.

What is the value of x ?

Show Answer

The correct answer is 97. The intersecting lines form a triangle, and the angle with measure of x degrees is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of 23 degrees and the other, which is supplementary to the angle with measure 106 degrees, has measure of 180 degrees minus 106 degrees, equals 74 degrees. Therefore, the value of x is 23 plus 74, equals 97.

Question 7 7 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled t, m, and n.
  • Line t intersects both line m and line n.
  • At the intersection of line t and line m, 1 angle is labeled clockwise from top left as follows:
    • Bottom left: 170°
  • At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:
    • Bottom left: w°
  • A note indicates the figure is not drawn to scale.

In the figure, line m is parallel to line n . What is the value of w ?

  1. 17

  2. 30

  3. 70

  4. 170

Show Answer Correct Answer: D

Choice D is correct. It's given that lines m and n are parallel. Since line t intersects both lines m and n , it's a transversal. The angles in the figure marked as 170° and w° are on the same side of the transversal, where one is an interior angle with line m as a side, and the other is an exterior angle with line n as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that w°=170°. Therefore, the value of w is 170

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 8 8 of 269 selected Area And Volume M

A right circular cylinder has a volume of 45 pi. If the height of the cylinder is 5, what is the radius of the cylinder?

  1. 3

  2. 4.5

  3. 9

  4. 40

Show Answer Correct Answer: A

Choice A is correct. The volume of a right circular cylinder with a radius of r is the product of the area of the base, pi, r squared, and the height, h. The volume of the right circular cylinder described is 45 pi and its height is 5. If the radius is r, it follows that 45 pi equals, pi times, r, squared, times 5. Dividing both sides of this equation by 5 pi yields 9 equals r squared. Taking the square root of both sides yields r equals 3 or r equals negative 3. Since r represents the radius, the value must be positive. Therefore, the radius is 3.

Choice B is incorrect and may result from finding that the square of the radius is 9, but then from dividing 9 by 2, rather than taking the square root of 9. Choice C is incorrect. This represents the square of the radius. Choice D is incorrect and may result from solving the equation 45 pi equals, pi times, r, squared, times 5 for r squared, not r, by dividing by pi on both sides and then by subtracting, not dividing, 5 from both sides.

 

Question 9 9 of 269 selected Area And Volume H

Square A has side lengths that are 166 times the side lengths of square B. The area of square A is k times the area of square B. What is the value of k ?

Show Answer Correct Answer: 27556

The correct answer is 27,556 . The area of a square is s 2 , where s is the side length of the square. Let x represent the length of each side of square B. Substituting x for s in s 2 yields x 2 . It follows that the area of square B is x 2 . It’s given that square A has side lengths that are 166 times the side lengths of square B. Since x represents the length of each side of square B, the length of each side of square A can be represented by the expression 166 x . It follows that the area of square A is (166x)2, or 27,556 x 2 . It’s given that the area of square A is k times the area of square B. Since the area of square A is equal to 27,556 x 2 , and the area of square B is equal to x 2 , an equation representing the given statement is 27,556 x 2 = k x 2 . Since x represents the length of each side of square B, the value of x must be positive. Therefore, the value of x 2 is also positive, so it does not equal 0 . Dividing by x 2 on both sides of the equation 27,556 x 2 = k x 2 yields 27,556=k. Therefore, the value of k is 27,556 .

Question 10 10 of 269 selected Right Triangles And Trigonometry M

In right triangle R S T , the sum of the measures of angle R and angle S is 90 degrees. The value of sin(R) is 15 4 . What is the value of cos(S)?

  1. 15 15

  2. 15 4

  3. 4 15 15

  4. 15

Show Answer Correct Answer: B

Choice B is correct. The sine of any acute angle is equal to the cosine of its complement. It’s given that in right triangle R S T , the sum of the measures of angle R and angle S is 90 degrees. Therefore, angle R and angle S are complementary, and the value of sinR is equal to the value of cosS. It's given that the value of sinR is 154, so the value of cosS is also 154.

Choice A is incorrect. This is the value of tanS.

Choice C is incorrect. This is the value of 1cosS.

Choice D is incorrect. This is the value of 1tanS.

Question 11 11 of 269 selected Area And Volume E

What is the area, in square inches, of a rectangle with a length of 7 inches and a width of 6 inches?

  1. 13

  2. 20

  3. 42

  4. 84

Show Answer Correct Answer: C

Choice C is correct. The area, A , of a rectangle is given by the formula A=lw, where l represents the length of the rectangle and w represents its width. It’s given that the rectangle has a length of 7 inches and a width of 6 inches. Substituting 7 for l and 6 for w in the formula A=lw yields A=(7)(6), or A = 42 . Thus, the area, in square inches, of the rectangle is 42 .

Choice A is incorrect. This is the sum, not the product, of the length and width of the rectangle.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is twice the area, in square inches, of the rectangle.

Question 12 12 of 269 selected Area And Volume E

The side length of a square is 55 centimeters (cm). What is the area, in cm2, of the square?

  1. 110

  2. 220

  3. 3,025

  4. 12,100

Show Answer Correct Answer: C

Choice C is correct. The area A , in square centimeters (cm2), of a square with side length s , in cm, is given by the formula A = s 2 . It’s given that the square has a side length of 55 cm. Substituting 55 for s in the formula A = s 2 yields A=552, or A = 3,025 . Therefore, the area, in cm2, of the square is 3,025 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the perimeter, in cm, of the square, not its area, in cm2.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 13 13 of 269 selected Area And Volume E

Triangle R has an area of 80 square centimeters (cm2). Square S has side lengths of 4 cm. What is the total area of triangle R and square S, in cm2?

  1. 42

  2. 44

  3. 84

  4. 96

Show Answer Correct Answer: D

Choice D is correct. It’s given that triangle R has an area of 80 cm2. The area of a square is l2, where l is the side length of the square. It's given that square S has side lengths of 4 cm. It follows that the area, in cm2, of square S is 42, or 16. Therefore, the total area, in cm2, of triangle R and square S is 80+16, or 96.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 14 14 of 269 selected Area And Volume M

What is the length of one side of a square that has the same area as a circle with radius 2 ?

  1. 2

  2. the square root of 2 pi, end root

  3. 2 times the square root of pi

  4. 2 pi

Show Answer Correct Answer: C

Choice C is correct. The area A of a circle with radius r is given by the formula A equals, pi times r squared. Thus, a circle with radius 2 has area pi times 2 squared, which can be rewritten as 4 pi. The area of a square with side length s is given by the formula A equals s squared. Thus, if a square has the same area as a circle with radius 2, then s squared equals 4 pi. Since the side length of a square must be a positive number, taking the square root of both sides of s squared equals 4 pi gives s equals, the square root of 4 pi, end root. Using the properties of square roots, the square root of 4 pi, end root can be rewritten as open parenthesis, the square root of 4, close parenthesis, times, open parenthesis, the square root of pi, close parenthesis, which is equivalent to 2 times the square root of pi. Therefore, s equals, 2 times the square root of pi.

Choice A is incorrect. The side length of the square isn’t equal to the radius of the circle. Choices B and D are incorrect and may result from incorrectly simplifying the expression the square root of 4 pi, end root.

 

Question 15 15 of 269 selected Area And Volume H

A right square prism has a height of 14 units. The volume of the prism is 2,016 cubic units. What is the length, in units, of an edge of the base?

Show Answer Correct Answer: 12

The correct answer is 12 . The volume, V , of a right square prism can be calculated using the formula V=s2h, where s represents the length of an edge of the base and h represents the height of the prism. It’s given that the volume of the prism is 2,016 cubic units and the height is 14 units. Substituting 2,016 for V and 14 for h in the formula V=s2h yields 2,016=(s2)(14). Dividing both sides of this equation by 14 yields 144 = s 2 . Taking the square root of both sides of this equation yields two solutions: - 12 = s and 12 = s . The length can't be negative, so 12 = s . Therefore, the length, in units, of an edge of the base is 12 .

Question 16 16 of 269 selected Right Triangles And Trigonometry H

In triangle X Y Z , angle Z is a right angle and the length of YZ¯ is 24 units. If tanX= 12 35 , what is the perimeter, in units, of triangle X Y Z ?

  1. 188

  2. 168

  3. 84

  4. 71

Show Answer Correct Answer: B

Choice B is correct. It's given that angle Z in triangle XYZ is a right angle. Thus, side YZ is the leg opposite angle X and side XZ is the leg adjacent to angle X . The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. It follows that tanX=YZXZ. It's given that tanX=1235 and the length of side YZ is 24 units. Substituting 1235 for tanX and 24 for YZ in the equation tanX=YZXZ yields 1235=24XZ. Multiplying both sides of this equation by 35(XZ) yields 12(XZ)=24(35), or 12(XZ)=840. Dividing both sides of this equation by 12 yields XZ=70. The length XY can be calculated using the Pythagorean theorem, which states that if a right triangle has legs with lengths of a and b and a hypotenuse with length c , then a2+b2=c2. Substituting 70 for a and 24 for b in this equation yields 702+242=c2, or 5,476=c2. Taking the square root of both sides of this equation yields ±74=c. Since the length of the hypotenuse must be positive, 74 = c . Therefore, the length of XY is 74 units. The perimeter of a triangle is the sum of the lengths of all sides. Thus, (74+70+24) units, or 168 units, is the perimeter of triangle XYZ.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This would be the perimeter, in units, for a right triangle where the length of side YZ is 12 units, not 24 units.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 17 17 of 269 selected Right Triangles And Trigonometry M

  • Angle upper Z is a right angle.
  • The length of side upper X upper Z is 27.
  • The length of side upper Y upper Z is 35.
  • A note indicates the figure is not drawn to scale.

Triangle X Y Z shown is a right triangle. Which of the following has the same value as sinX?

  1. tanX

  2. tanY

  3. cosX

  4. cosY

Show Answer Correct Answer: D

Choice D is correct. The sine of an angle is equal to the cosine of its complementary angle. In the triangle shown, angle Z is a right angle; thus, angles X and Y are complementary angles. Therefore, cosY has the same value as sinX.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 18 18 of 269 selected Lines, Angles, And Triangles E
The figure presents triangle B A, C, such that side B C is horizontal and vertex A, is above side B C. Angle A, is labeled x degrees.

In the given triangle, the length of side A, B equals the length of side A, C and angle A, B C has a measure of 67 degrees. What is the value of x ?

  1. 36

  2. 46

  3. 58

  4. 70

Show Answer Correct Answer: B

Choice B is correct. Since the length of A, B equals the length of A, C, the measures of their corresponding angles, angle A, B C and angle A, C B, are equal. Since angle A, B C has a measure of 67 degrees, the measure of angle A, C B is also 67 degrees. Since the sum of the measures of the interior angles in a triangle is 180 degrees, it follows that 67 plus 67, plus x, equals 180, or 134 plus x, equals 180. Subtracting by 134 on both sides of this equation yields x equals 46.

Choices A, C, and D are incorrect and may result from calculation errors.

Question 19 19 of 269 selected Lines, Angles, And Triangles H
The figure presents line segments MQ and NR that intersect at point P, where point N is above and slightly to the right of point M and point Q is above and slightly to the left of point R. Line segments MN, QR, and horizontal line segment M, R are drawn forming triangles MNR and QMR. The measure of angle QPR is labeled 60 degrees and the angle measure of MQR is labeled 70 degrees.

In the figure above, line segment M Q and line segment N R intersect at point P, N P equals Q P, and M P equals P R. What is the measure, in degrees, of angle Q M R ? (Disregard the degree symbol when gridding your answer.)

Show Answer

The correct answer is 30. It is given that the measure of angle Q P R is 60 degrees. Angle MPR and angle Q P R are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is 180 degrees, and so the measure of angle M P R is 120 degrees. The sum of the angles in a triangle is 180 degrees. Subtracting the measure of angle M P R from 180 degrees yields the sum of the other angles in the triangle MPR. Since 180 minus 120, equals 60, the sum of the measures of angle Q M R and angle N R M is 60 degrees. It is given that the length of side M P equals the length of side P R, so it follows that triangle MPR is isosceles. Thereforeangle Q M R and angle N R M must be congruent. Since the sum of the measure of these two angles is 60 degrees, it follows that the measure of each angle is 30 degrees.

An alternate approach would be to use the exterior angle theorem, noting that the measure of angle Q P R is equal to the sum of the measures of angle Q M R and angle N R M. Since both angles are equal, each of them has a measure of 30 degrees.

Question 20 20 of 269 selected Area And Volume M

Square X has a side length of 12 centimeters. The perimeter of square Y is 2 times the perimeter of square X. What is the length, in centimeters, of one side of square Y?

  1. 6

  2. 10

  3. 14

  4. 24

Show Answer Correct Answer: D

Choice D is correct. The perimeter, P , of a square can be found using the formula P = 4 s , where s is the length of each side of the square. It's given that square X has a side length of 12 centimeters. Substituting 12 for s in the formula for the perimeter of a square yields P=4(12), or P = 48 . Therefore, the perimeter of square X is 48 centimeters. It’s also given that the perimeter of square Y is 2 times the perimeter of square X. Therefore, the perimeter of square Y is 2(48), or 96 , centimeters. Substituting 96 for P in the formula P=4s gives 96=4s. Dividing both sides of this equation by 4 gives 24 = s . Therefore, the length of one side of square Y is 24 centimeters.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 21 21 of 269 selected Area And Volume H

The circumference of the base of a right circular cylinder is 20π meters, and the height of the cylinder is 6 meters. What is the volume, in cubic meters, of the cylinder?

  1. 60π

  2. 120π

  3. 600π

  4. 2,400π

Show Answer Correct Answer: C

Choice C is correct. The volume, V, of a right circular cylinder is given by the formula V=πr2h, where r is the radius of the base of the cylinder and h is the height of the cylinder. It’s given that a right circular cylinder has a height of 6 meters. Therefore, h=6. It's also given that the right circular cylinder has a base with a circumference of 20π meters. The circumference, C, of a circle is given by C=2πr, where r is the radius of the circle. Substituting 20π for C in the formula C=2πr yields 20π=2πr. Dividing each side of this equation by 2π yields 10=r. Substituting 10 for r and 6 for h in the formula V=πr2h yields V=π(10)2(6), or V=600π. Therefore, the volume, in cubic meters, of the cylinder is 600π.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the lateral surface area, not the volume, of the cylinder.

Choice D is incorrect. This is the result of using the diameter, not the radius, for the value of r in the formula V=πr2h.

Question 22 22 of 269 selected Lines, Angles, And Triangles E

  • Right triangle upper P upper Q upper R is labeled as follows:
    • Angle upper R is a right angle.
  • Right triangle upper S upper T upper U is labeled as follows:
    • Angle upper U is a right angle.
  • A note indicates the figures are not drawn to scale.

Right triangles P Q R and S T U are similar, where P corresponds to S . If the measure of angle Q is 18°, what is the measure of angle S ?

  1. 18°

  2. 72°

  3. 82°

  4. 162°

Show Answer Correct Answer: B

Choice B is correct. In similar triangles, corresponding angles are congruent. It’s given that right triangles P Q R and S T U are similar, where angle P corresponds to angle S . It follows that angle P is congruent to angle S . In the triangles shown, angle R and angle U are both marked as right angles, so angle R and angle U are corresponding angles. It follows that angle Q and angle T are corresponding angles, and thus, angle Q is congruent to angle T . It’s given that the measure of angle Q is 18°, so the measure of angle T is also 18°. Angle U is a right angle, so the measure of angle U is 90°. The sum of the measures of the interior angles of a triangle is 180°. Thus, the sum of the measures of the interior angles of triangle S T U is 180 degrees. Let s represent the measure, in degrees, of angle S . It follows that s+18+90=180, or s + 108 = 180 . Subtracting 108 from both sides of this equation yields s = 72 . Therefore, if the measure of angle Q is 18 degrees, then the measure of angle S is 72 degrees.

Choice A is incorrect. This is the measure of angle T .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the sum of the measures of angle S and angle U .

Question 23 23 of 269 selected Area And Volume E

  • Right triangle upper J upper K upper L is labeled as follows:
    • Angle upper L is a right angle.
  • Right triangle upper R upper S upper T is labeled as follows:
    • Angle upper T is a right angle.
  • A note indicates the figures are not drawn to scale.

 

In the figure shown, triangle J K L is similar to triangle R S T , where J corresponds to R and K corresponds to S . The length of JK¯ is 15 , and the perimeter of triangle J K L is 36 . The length of RS¯ is 135 . What is the perimeter of triangle R S T ?

  1. 324

  2. 540

  3. 2,916

  4. 8,100

Show Answer Correct Answer: A

Choice A is correct. It’s given that triangle JKL is similar to triangle RST, where J corresponds to R and K corresponds to S. It follows that JK¯ corresponds to RS¯. If two triangles are similar, then the scale factor between their perimeters is equal to the scale factor between the lengths of their corresponding sides. It's given that the length of JK¯ is 15 and the length of RS¯ is 135. Therefore, the scale factor from the length of JK¯ to the length of RS¯ is 13515, or 9. It’s given that the perimeter of triangle JKL is 36. Let p represent the perimeter of triangle RST. It follows that p36=9. Multiplying each side of this equation by 36 yields p=324. Therefore, the perimeter of triangle RST is 324.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 24 24 of 269 selected Lines, Angles, And Triangles M

  • Clockwise from top left, the 3 lines are labeled s, q, and r.
  • Line s intersects both line q and line r.
  • At the intersection of line s and line q, 1 angle is labeled clockwise from top left as follows:
    • Top right: 58°
  • At the intersection of line s and line r, 1 angle is labeled clockwise from top left as follows:
    • Top left: y°
  • A note indicates the figure is not drawn to scale.

In the figure, line q is parallel to line r , and both lines are intersected by line s . If y = 2 x + 8 , what is the value of x ?

Show Answer Correct Answer: 57

The correct answer is 57 . Based on the figure, the angle with measure y° and the angle vertical to the angle with measure 58° are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure 58° also has measure 58°. It’s given that lines q and r are parallel. Therefore, same side interior angles between lines q and r are supplementary. It follows that y + 58 = 180 . If y = 2 x + 8 , then the value of x can be found by substituting 2 x + 8 for y in the equation y + 58 = 180 , which yields (2x+8)+58=180, or 2x+66=180. Subtracting 66 from both sides of this equation yields 2 x = 114 . Dividing both sides of this equation by 2 yields x = 57 . Thus, if y = 2 x + 8 , the value of x is 57 .

Question 25 25 of 269 selected Lines, Angles, And Triangles E

The figure presents 2 parallel lines l and m, which are almost horizontal, with l above m. A line segment from the left on m, moves up and to the right, intersects, and ends above l. Another line segment from the right on m, moves up and to the left, intersects, and ends at the same point as the other line segment, above l. The angle at the point where the two line segments meet is labeled x degrees. The angle above l and to the left of the right line segment is labeled z degrees. The angle above m and to the right of the left line segment is labeled y degrees. The figure is not drawn to scale

In the figure above, lines l and m are parallel, y equals 20, and z equals 60. What is the value of x ?

  1. 120

  2. 100

  3. 90

  4. 80

Show Answer Correct Answer: B

Choice B is correct. Let the measure of the third angle in the smaller triangle be a, degrees. Since lines l and m are parallel and cut by transversals, it follows that the corresponding angles formed are congruent. So a degrees, equals y degrees, which equals 20 degrees. The sum of the measures of the interior angles of a triangle is 180 degrees, which for the interior angles in the smaller triangle yields a, plus x, plus z, equals 180. Given that z equals 60 and a, equals 20, it follows that 20 plus x, plus 60, equals 180. Solving for x gives x equals, 180 minus 60, minus 20, or x equals 100.

Choice A is incorrect and may result from incorrectly assuming that angles x plus z, equals 180. Choice C is incorrect and may result from incorrectly assuming that the smaller triangle is a right triangle, with x as the right angle. Choice D is incorrect and may result from a misunderstanding of the exterior angle theorem and incorrectly assuming that x equals, y plus z.

Question 26 26 of 269 selected Lines, Angles, And Triangles M

The figure presents quadrilateral A, C D F, where side A, F and side C D are horizontal, and side C D lies above side A, F. Side C D is longer than side A, F. Point B lies on left side A, C, and point E lies on right side D F. Line segment B E, which is horizontal, divides quadrilateral A, C D F into two quadrilaterals, A, B E F and B C D E.

In the figure above, side A, F , line segment B E , and side C D are parallel. Points B and E lie on side A, C  and side F D , respectively. If the length of line segment A, B equals 9, the length of line segment B C equals 18 point 5, and the length of line segment F E equals 8 point 5, what is the length of line segment E D , to the nearest tenth?

  1. 16.8

  2. 17.5

  3. 18.4

  4. 19.6

Show Answer Correct Answer: B

Choice B is correct. Since line segment A F, line segment B E, and line segment C D are parallel, quadrilaterals A F E B and B E D C are similar. Let x represent the length of line segment E D. With similar figures, the ratios of the lengths of corresponding sides are equal. It follows that 9 over 18 point 5, equals 8 point 5 over x. Multiplying both sides of this equation by 18.5 and by x yields 9 x equals, 18 point 5, times 8 point 5, or 9 x equals 157 point 2 5. Dividing both sides of this equation by 9 yields x equals 17 point 4 7, which to the nearest tenth is 17.5.

Choices A, C, and D are incorrect and may result from errors made when setting up the proportion.

Question 27 27 of 269 selected Area And Volume E

The area of a square is 64 square inches. What is the side length, in inches, of this square?

  1. 8

  2. 16

  3. 64

  4. 128

Show Answer Correct Answer: A

Choice A is correct. It's given that the area of a square is 64 square inches. The area A , in square inches, of a square is given by the formula A=s2, where s is the side length, in inches, of the square. Substituting 64 for A in this formula yields 64=s2. Taking the positive square root of both sides of this equation yields 8=s. Thus, the side length, in inches, of this square is 8 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the area, in square inches, of the square, not the side length, in inches, of the square.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 28 28 of 269 selected Right Triangles And Trigonometry M
The figure presents right triangle A B C. Side A C is horizontal, with A to the left of C, and side B C is vertical, with B above C. A right angle symbol is indicated at angle C. Side A B is labeled 29. Side A C is labeled 21. Side B C is labeled 20.

In the figure above, what is the value of tangent A?

  1. 20 over 29

  2. 21 over 29

  3. 20 over 21

  4. 21 over 20

Show Answer Correct Answer: C

Choice C is correct. Angle A is an acute angle in a right triangle, so the value of tan(A) is equivalent to the ratio of the length of the side opposite angle A, 20, to the length of the nonhypotenuse side adjacent to angle A, 21. Therefore,  tangent A, equals 20 over 21.

Choice A is incorrect. This is the value of sin(A). Choice B is incorrect. This is the value of cos(A). Choice D is incorrect. This is the value of tan(B).

 

Question 29 29 of 269 selected Area And Volume M

A circle has a radius of 43 meters. What is the area, in square meters, of the circle?

  1. 43 π 2

  2. 43 π

  3. 86 π

  4. 1,849 π

Show Answer Correct Answer: D

Choice D is correct. The area, A , of a circle is given by the formula A=πr2, where r is the radius of the circle. It’s given that the circle has a radius of 43 meters. Substituting 43 for r in the formula A=πr2 yields A=π(43)2, or A=1,849π. Therefore, the area, in square meters, of the circle is 1,849π.

Choice A is incorrect. This is the area, in square meters, of a circle with a radius of 432 meters.

Choice B is incorrect. This is the area, in square meters, of a circle with a radius of 43 meters.

Choice C is incorrect. This is the circumference, in meters, of the circle.

Question 30 30 of 269 selected Circles H

A circle in the xy-plane has its center at (-1,1). Line t is tangent to this circle at the point (5,-4). Which of the following points also lies on line t ?

  1. (0,65)

  2. (4,7)

  3. (10,2)

  4. (11,1)

Show Answer Correct Answer: C

Choice C is correct. It’s given that the circle has its center at (-1,1) and that line t is tangent to this circle at the point (5,-4). Therefore, the points (-1,1) and (5,-4) are the endpoints of the radius of the circle at the point of tangency. The slope of a line or line segment that contains the points (a,b) and (c,d) can be calculated as d-bc-a. Substituting (-1,1) for (a,b) and (5,-4) for (c,d) in the expression d-bc-a yields -4-15-(-1) , or -56. Thus, the slope of this radius is -56. A line that’s tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It follows that line t is perpendicular to the radius at the point (5,-4), so the slope of line t is the negative reciprocal of the slope of this radius. The negative reciprocal of -56 is 65. Therefore, the slope of line t is 65. Since the slope of line t is the same between any two points on line t , a point lies on line t if the slope of the line segment connecting the point and (5,-4) is 65. Substituting choice C, (10,2), for (a,b) and (5,-4) for (c,d) in the expression d-bc-a yields -4-25-10, or 65. Therefore, the point (10,2) lies on line t .

Choice A is incorrect. The slope of the line segment connecting (0,65) and (5,-4) is -4-655-0, or -2625, not 65.

Choice B is incorrect. The slope of the line segment connecting (4,7) and (5,-4) is -4-75-4, or -11 , not 65.

Choice D is incorrect. The slope of the line segment connecting (11,1) and (5,-4) is -4-15-11, or 56, not 65.

Question 31 31 of 269 selected Circles H

Points Q and R lie on a circle with center P . The radius of this circle is 9 inches. Triangle P Q R has a perimeter of 31 inches. What is the length, in inches, of QR¯?

  1. 132

  2. 13

  3. 92

  4. 9

Show Answer Correct Answer: B

Choice B is correct. Since it's given that P is the center of a circle with a radius of 9 inches, and that points Q and R lie on that circle, it follows that PQ¯ and RP¯ of triangle PQR each have a length of 9 inches. Let the length of QR¯ be x inches. It follows that the perimeter of triangle PQR is 9+9+x inches. Since it's given that the perimeter of triangle PQR is 31 inches, it follows that 9+9+x=31, or 18+x=31. Subtracting 18 from both sides of this equation gives x = 13 . Therefore, the length, in inches, of QR¯ is 13 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 32 32 of 269 selected Area And Volume M

  • From left to right, the 3 points have the following coordinates:
    • (4 comma 8)
    • (7 comma 11)
    • (7 comma 5)

The three points shown define a circle. The circumference of this circle is kπ, where k is a constant. What is the value of k ?

  1. 3

  2. 6

  3. 7

  4. 9

Show Answer Correct Answer: B

Choice B is correct. It’s given that the three points shown define a circle, so the center of that circle is an equal distance from each of the three points. The point (7,8) is halfway between the points (7,5) and (7,11) and is a distance of 3 units from each of those two points. The point (7,8) is also a distance of 3 units from (4,8). Because the point (7,8) is the same distance from all three given points, it must be the center of the circle. The radius of a circle is the distance from the center to any point on the circle. Since that distance is 3, it follows that the radius of the circle is 3. The circumference of a circle with radius r is equal to 2πr. It follows that the circumference of the circle is 2(π)(3), or 6π. It's given that the circumference of the circle is kπ. Therefore, the value of k is 6.

Choice A is incorrect. This is the radius of the circle, not the value of k in the expression kπ.

Choice C is incorrect. This is the x-coordinate of the center of the circle, not the value of k in the expression kπ.

Choice D is incorrect. This is the value of k for which kπ represents the area of the circle, in square units, not the circumference of the circle, in units.

Question 33 33 of 269 selected Area And Volume M

What is the volume, in cubic centimeters, of a right rectangular prism that has a length of 4 centimeters, a width of 9 centimeters, and a height of 10 centimeters?

Show Answer

The correct answer is 360. The volume of a right rectangular prism is calculated by multiplying its dimensions: length, width, and height. Multiplying the values given for these dimensions yields a volume of 4 times 9, times 10, equals 360 cubic centimeters.

Question 34 34 of 269 selected Area And Volume H

A cube has an edge length of 68 inches. A solid sphere with a radius of 34 inches is inside the cube, such that the sphere touches the center of each face of the cube. To the nearest cubic inch, what is the volume of the space in the cube not taken up by the sphere?

  1. 149,796

  2. 164,500

  3. 190,955

  4. 310,800

Show Answer Correct Answer: A

Choice A is correct. The volume of a cube can be found by using the formula V=s3, where V is the volume and s is the edge length of the cube. Therefore, the volume of the given cube is V=683, or 314,432 cubic inches. The volume of a sphere can be found by using the formula V=43πr3 , where V is the volume and r is the radius of the sphere. Therefore, the volume of the given sphere is V=43π(34)3, or approximately 164,636 cubic inches. The volume of the space in the cube not taken up by the sphere is the difference between the volume of the cube and volume of the sphere. Subtracting the approximate volume of the sphere from the volume of the cube gives 314,432-164,636=149,796 cubic inches.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 35 35 of 269 selected Area And Volume E

A right rectangular prism has a length of 11 meters, a width of 8 meters, and a height of 10 meters. What is the volume, in cubic meters, of the prism?

Show Answer Correct Answer: 880

The correct answer is 880. The volume, V, of a right rectangular prism is given by the formula V=lwh, where l is the length, w is the width, and h is the height of the prism. It’s given that a right rectangular prism has a length of 11 meters, a width of 8 meters, and a height of 10 meters. Substituting 11 for l, 8 for w, and 10 for h in the formula V=lwh yields V=(11)(8)(10), or V=880. Therefore, the volume, in cubic meters, of the prism is 880.

Question 36 36 of 269 selected Lines, Angles, And Triangles H

In triangle A B C , the measure of angle B is 90° and BD is an altitude of the triangle. The length of AB is 15 and the length of AC is 23 greater than the length of AB. What is the value of BCBD?

  1. 15 38

  2. 15 23

  3. 23 15

  4. 38 15

Show Answer Correct Answer: D

Choice D is correct. It's given that in triangle ABC, the measure of angle B is 90° and  BD is an altitude of the triangle. Therefore, the measure of angle BDC is 90°. It follows that angle B is congruent to angle D and angle C is congruent to angle C . By the angle-angle similarity postulate, triangle ABC is similar to triangle BDC. Since triangles ABC and BDC are similar, it follows that ACAB=BCBD. It's also given that the length of AB¯ is 15 and the length of AC¯ is 23 greater than the length of AB¯. Therefore, the length of AC¯ is 15+23, or 38 . Substituting 15 for AB and 38 for AC in the equation ACAB=BCBD yields 3815=BCBD. Therefore, the value of BCBD is 3815.

Choice A is incorrect. This is the value of BDBC.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.
 

Question 37 37 of 269 selected Area And Volume H

The figure shown is a right circular cylinder with a radius of r and height of h . A second right circular cylinder (not shown) has a volume that is 392 times as large as the volume of the cylinder shown. Which of the following could represent the radius R , in terms of r , and the height H , in terms of h , of the second cylinder?

  1. R = 8 r and H = 7 h

  2. R = 8 r and H = 49 h

  3. R = 7 r and H = 8 h

  4. R = 49 r and H = 8 h

Show Answer Correct Answer: C

Choice C is correct. The volume of a right circular cylinder is equal to πa2b, where a is the radius of a base of the cylinder and b is the height of the cylinder. It’s given that the cylinder shown has a radius of r and a height of h . It follows that the volume of the cylinder shown is equal to πr2h. It’s given that the second right circular cylinder has a radius of R and a height of H . It follows that the volume of the second cylinder is equal to πR2H. Choice C gives R = 7 r and H = 8 h . Substituting 7 r for R and 8 h for H in the expression that represents the volume of the second cylinder yields π(7r)2(8h), or π(49r2)(8h), which is equivalent to π(392r2h), or 392(πr2h). This expression is equal to 392 times the volume of the cylinder shown, πr2h. Therefore, R = 7 r and H = 8 h could represent the radius R , in terms of r , and the height H , in terms of h , of the second cylinder.

Choice A is incorrect. Substituting 8 r for R and 7 h for H in the expression that represents the volume of the second cylinder yields π(8r)2(7h), or π(64r2)(7h), which is equivalent to π(448r2h), or 448(πr2h). This expression is equal to 448 , not 392 , times the volume of the cylinder shown. 

Choice B is incorrect. Substituting 8 r for R and 49 h for H in the expression that represents the volume of the second cylinder yields π(8r)2(49h), or π(64r2)(49h), which is equivalent to π(3,136r2h), or 3,136(πr2h). This expression is equal to 3,136 , not 392 , times the volume of the cylinder shown.

Choice D is incorrect. Substituting 49 r for R and 8 h for H in the expression that represents the volume of the second cylinder yields π(49r)2(8h), or π(2,401r2)(8h), which is equivalent to π(19,208r2h), or 19,208(πr2h). This expression is equal to 19,208 , not 392 , times the volume of the cylinder shown.

Question 38 38 of 269 selected Lines, Angles, And Triangles E

In XYZ, the measure of X is 24 ° and the measure of Y is 98 °. What is the measure of Z?

  1. 58 °

  2. 74 °

  3. 122 °

  4. 212 °

Show Answer Correct Answer: A

Choice A is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is 180°. It's given that in XYZ, the measure of X is 24° and the measure of Y is 98°. It follows that the measure of Z is (180-24-98)°, or 58°.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the sum of the measures of X and Y, not the measure of Z.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 39 39 of 269 selected Area And Volume M

A right circular cylinder has a volume of 377 cubic centimeters. The area of the base of the cylinder is 13 square centimeters. What is the height, in centimeters, of the cylinder?

Show Answer Correct Answer: 29

The correct answer is 29 . The volume, V , of a right circular cylinder is given by the formula V=πr2h, where r is the radius of the base of the cylinder and h is the height of the cylinder. Since the base of the cylinder is a circle with radius r , the area of the base of the cylinder is πr2. It's given that a right circular cylinder has a volume of 377 cubic centimeters; therefore, V = 377 . It's also given that the area of the base of the cylinder is 13 square centimeters; therefore, πr2=13. Substituting 377 for V and 13 for πr2 in the formula V=πr2h yields 377 = 13 h . Dividing both sides of this equation by 13 yields 29 = h . Therefore, the height of the cylinder, in centimeters, is 29 .

Question 40 40 of 269 selected Lines, Angles, And Triangles M

Two nearby trees are perpendicular to the ground, which is flat. One of these trees is 10 feet tall and has a shadow that is 5 feet long. At the same time, the shadow of the other tree is 2 feet long. How tall, in feet, is the other tree?

  1. 3

  2. 4

  3. 8

  4. 27

Show Answer Correct Answer: B

Choice B is correct. Each tree and its shadow can be modeled using a right triangle, where the height of the tree and the length of its shadow are the legs of the triangle. At a given point in time, the right triangles formed by two nearby trees and their respective shadows will be similar. Therefore, if the height of the other tree is x , in feet, the value of x can be calculated by solving the proportional relationship 10 feet tall5 feet long=x feet tall2 feet long. This equation is equivalent to105=x2, or 2=x2. Multiplying each side of the equation 2=x2 by 2 yields 4=x. Therefore, the other tree is 4 feet tall.

Choice A is incorrect and may result from calculating the difference between the lengths of the shadows, rather than the height of the other tree.

Choice C is incorrect and may result from calculating the difference between the height of the 10 -foot-tall tree and the length of the shadow of the other tree, rather than calculating the height of the other tree.

Choice D is incorrect and may result from a conceptual or calculation error.

Question 41 41 of 269 selected Lines, Angles, And Triangles M

  • The 2 triangles share common vertex upper R.
    • Common vertex upper R lies on line segment upper P upper T.
  • Triangle upper Q upper P upper R is labeled as follows:
    • Angle upper Q upper P upper R is a right angle.
    • The measure of angle upper Q upper R upper P is x°.
  • Triangle upper S upper T upper R is labeled as follows:
    • Angle upper S upper T upper R is a right angle.
    • The measure of angle upper S upper R upper T is x°.
  • A note indicates the figure is not drawn to scale.

 

QPR is similar to STR. The lengths represented by ST¯QP¯PR¯, and QR¯ in the figure are 14 , 15 , 20 , and 25 , respectively. What is the length of SR¯?

  1. 35015

  2. 35020

  3. 21020

  4. 21025

Show Answer Correct Answer: A

Choice A is correct. The figure shows that angle P in QPR and angle T in STR are right angles. It follows that angle P is congruent to angle T . The figure also shows that the measures of angle QRP and angle SRT are both x°. Therefore, angle QRP is congruent to angle SRT. It’s given that QPR is similar to STR. Since angle P is congruent to angle T , and angle QRP is congruent to angle SRT, it follows that QR¯ corresponds to SR¯, and QP¯ corresponds to ST¯. Since corresponding sides of similar triangles are proportional, it follows that SRQR=STQP. It’s also given that the lengths of ST¯, QP¯, and QR¯ are 14 , 15 , and 25 , respectively. Substituting 14 for S T , 15 for QP, and 25 for Q R in the equation SRQR=STQP yields SR25=1415. Multiplying each side of this equation by 25 yields SR=(1415)(25), or SR=35015. Thus, the length of SR¯ is 35015.

Choice B is incorrect. This is the result of solving the equation SR25=1420, not SR25=1415.

Choice C is incorrect. This is the result of solving the equation SR14=1520, not SR25=1415.

Choice D is incorrect. This is the result of solving the equation SR14=1525, not SR25=1415.

Question 42 42 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled t, m, and n.
  • Line t intersects both line m and line n.
  • At the intersection of line t and line m, 1 angle is labeled clockwise from top left as follows:
    • Bottom left: 134°
  • At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:
    • Bottom left: w°
  • A note indicates the figure is not drawn to scale. 

In the figure, line m is parallel to line n . What is the value of w ?

  1. 13

  2. 34

  3. 66

  4. 134

Show Answer Correct Answer: D

Choice D is correct. It's given that lines m and n are parallel. Since line t intersects both lines m and n , it's a transversal. The angles in the figure marked as 134° and w° are on the same side of the transversal, where one is an interior angle with line m as a side, and the other is an exterior angle with line n as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that w°=134°. Therefore, the value of w is 134 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 43 43 of 269 selected Lines, Angles, And Triangles M

Triangle A B C is similar to triangle X Y Z , where A , B , and C correspond to X , Y , and Z , respectively. In triangle A B C , the length of AB¯ is 170 and the length of BC¯ is 850 . In triangle X Y Z , the length of YZ¯ is 60 . What is the length of XY¯?

  1. 204

  2. 182

  3. 60

  4. 12

Show Answer Correct Answer: D

Choice D is correct. It's given that triangle A B C is similar to triangle X Y Z , where A , B , and C correspond to X , Y , and Z , respectively. It follows that side A B corresponds to side X Y and side B C corresponds to side Y Z . Since the lengths of corresponding sides in similar triangles are proportional, it follows that XYAB=YZBC. Substituting 170 for A B , 60 for Y Z , and 850 for B C in this equation yields XY170=60850. Multiplying each side of this equation by 170 yields X Y = 12 . Therefore, the length of XY¯ is 12 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the length of YZ¯, not XY¯.

Question 44 44 of 269 selected Lines, Angles, And Triangles H

  • Triangle upper R upper T upper U is labeled as follows:
    • The measure of angle upper R upper T upper U is x°.
  • The 2 triangles partially overlap:
    • Side upper R upper T of triangle upper R upper T upper U intersects side upper S upper V of triangle upper S upper V upper U.
    • Vertex upper V of triangle upper S upper V upper U lies on side upper R upper U of triangle upper R upper T upper U.
    • Vertex upper T of triangle upper R upper T upper U lies on side upper S upper U of triangle upper S upper V upper U.
  • A note indicates the figure is not drawn to scale.

 

In the figure, RT=TU, the measure of angle VST is 29°, and the measure of angle RVS is 41°. What is the value of x ?

Show Answer Correct Answer: 156

The correct answer is 156 . In the figure shown, the sum of the measures of angle UVS and angle RVS is 180°. It’s given that the measure of angle RVS is 41°. Therefore, the measure of angle UVS is (180-41)°, or 139°. The sum of the measures of the interior angles of a triangle is 180°. In triangle UVS, the measure of angle UVS is 139° and it's given that the measure of angle VST is 29°. Thus, the measure of angle VUS is (180-139-29)°, or 12°. It’s given that RT=TU. Therefore, triangle TUR is an isosceles triangle and the measure of VUS is equal to the measure of angle TRU. In triangle TUR, the measure of angle VUS is 12° and the measure of angle TRU is 12°. Thus, the measure of angle UTR is (180-12-12)°, or 156°. The figure shows that the measure of angle UTR is x°, so the value of x is 156 .

Question 45 45 of 269 selected Lines, Angles, And Triangles H

  • The 2 triangles intersect at vertex upper Q.
  • The measure of angle upper X upper W upper Q is a°.
  • The measure of angle upper Z upper Y upper Q is a°.
  • A note indicates the figure is not drawn to scale.

In the figure shown, WZ¯ and XY¯ intersect at point Q YQ=63WQ=70WX=60, and XQ=120. What is the length of YZ¯?

Show Answer Correct Answer: 54

The correct answer is 54 . The figure shown includes two triangles, triangle WQX and triangle YQZ, such that angle WQX and angle YQZ are vertical angles. It follows that angle WQX is congruent to angle YQZ. It’s also given in the figure that the measures of angle W and angle Y are a°. Therefore angle W is congruent to angle Y . Since triangle WQX and triangle YQZ have two pairs of congruent angles, triangle WQX is similar to triangle YQZ by the angle-angle similarity postulate, where YZ¯ corresponds to WX¯, and YQ¯ corresponds to WQ¯. Since the lengths of corresponding sides in similar triangles are proportional, it follows that YZWX=YQWQ. It’s given that YQ=63, WQ=70, and WX=60. Substituting 63 for YQ, 70 for WQ, and 60 for WX in the equation YZWX=YQWQ yields YZ60=6370. Multiplying each side of this equation by 60 yields YZ=(6370)(60), or YZ=54. Therefore, the length of YZ¯ is 54 .

Question 46 46 of 269 selected Circles H
The figure presents a circle with center O. There are four points on the circle. Going clockwise around the circle, the four points are A, B, C, and D.  Points A and C divide the circle into two arcs, arc A, D C and arc A B C. The length of arc A D C is less than the length of arc A, B C. Radius O A and radius O C are drawn. Angle A O C, the central angle corresponding to arc A, D C, measures x degrees

The circle above has center O, the length of arc A, D C is 5 pi, and x equals 100. What is the length of arc A, B C ?

  1. 9 pi

  2. 13 pi

  3. 18 pi

  4. 13 halves pi

Show Answer Correct Answer: B

Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It’s given that arc A D C is subtended by a central angle with measure 100°. Since the sum of the measures of the angles about a point is 360°, it follows that arc A B C is subtended by a central angle with measure 360 degrees minus 100 degrees, equals 260 degrees. If s is the length of arc A B C, then s must satisfy the ratio the fraction s over 5 pi, end fraction equals, the fraction 260 over 100. Reducing the fraction 260 over 100 to its simplest form gives the fraction 13 over 5. Therefore, the fraction s over 5 pi, end fraction, equals, the fraction 13 over 5. Multiplying both sides of the fraction s over 5 pi, end fraction, equals, the fraction 13 over 5 by 5 pi yields s equals 13 pi.

Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc A B C is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc A B C, not its full length.

 

Question 47 47 of 269 selected Lines, Angles, And Triangles H

In triangles A B C and D E F , angles B and E each have measure 27° and angles C and F each have measure 41°. Which additional piece of information is sufficient to determine whether triangle A B C is congruent to triangle D E F ?

  1. The measure of angle A

  2. The length of side A B

  3. The lengths of sides B C and E F

  4. No additional information is necessary.

Show Answer Correct Answer: C

Choice C is correct. Since angles B and E each have the same measure and angles C and F each have the same measure, triangles A B C and D E F are similar, where side B C corresponds to side E F . To determine whether two similar triangles are congruent, it is sufficient to determine whether one pair of corresponding sides are congruent. Therefore, to determine whether triangles A B C and D E F are congruent, it is sufficient to determine whether sides B C and E F have equal length. Thus, the lengths of B C and E F are sufficient to determine whether triangle A B C is congruent to triangle D E F .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect. The given information is sufficient to determine that triangles A B C and D E F are similar, but not whether they are congruent.

Question 48 48 of 269 selected Right Triangles And Trigonometry M

  • Angle upper A upper C upper B is a right angle.
  • The length of side upper A upper B is 42.
  • The length of side upper A upper C is 41.
  • A note indicates the figure is not drawn to scale.

What is the value of cosA in the triangle shown?

  1. 4241

  2. 4142

  3. 142

  4. 141

Show Answer Correct Answer: B

Choice B is correct. The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. In the triangle shown, the length of the leg adjacent to angle A is 41 , and the length of the hypotenuse is 42 . Therefore, cosA=4142.

Choice A is incorrect. This is the value of 1cosA.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 49 49 of 269 selected Right Triangles And Trigonometry H

In triangle X Y Z , angle Y is a right angle, the measure of angle Z is 33°, and the length of YZ¯ is 26 units. If the area, in square units, of triangle X Y Z can be represented by the expression ktan33°, where k is a constant, what is the value of k ?

Show Answer Correct Answer: 338

The correct answer is 338 . The tangent of an acute angle in a right triangle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. In triangle XYZ, it's given that angle Y is a right angle. Thus, XY¯ is the leg opposite of angle Z and YZ¯ is the leg adjacent to angle Z . It follows that tanZ=XYYZ. It's also given that the measure of angle Z is 33° and the length of YZ¯ is 26 units. Substituting 33° for Z and 26 for YZ in the equation tanZ=XYYZ yields tan33°=XY26. Multiplying each side of this equation by 26 yields 26tan33°=XY. Therefore, the length of XY¯ is 26tan33°. The area of a triangle is half the product of the lengths of its legs. Since the length of YZ¯ is 26 and the length of XY¯ is 26tan33°, it follows that the area of triangle XYZ is 12(26)(26tan33°) square units, or 338tan33° square units. It's given that the area, in square units, of triangle XYZ can be represented by the expression ktan33°, where k is a constant. Therefore, 338 is the value of k .

Question 50 50 of 269 selected Right Triangles And Trigonometry H
The figure presents right triangle A, B C. Side A, C is horizontal, and vertex B is directly above vertex C. Angle C is a right angle. The length of side A, B is 26.

Triangle A, B C above is a right triangle, and the sine of B, equals 5 over 13. What is the length of side B C?

Show Answer

The correct answer is 24. The sine of an acute angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the sine of angle B, or sine of B, is equal to the ratio of the length of side A, C to the length of side A, B. It’s given that the length of side A, B is 26 and that sine of B equals, 5 over 13. Therefore, the fraction 5 over 13, equals, the fraction A, B over 26. Multiplying both sides of this equation by 26 yields A, C equals 10.

By the Pythagorean Theorem, the relationship between the lengths of the sides of triangle ABC is as follows: 26 squared equals, 10 squared, plus B C squared, or 676 equals, 100 plus B C squared. Subtracting 100 from both sides of 676 equals, 100 plus B C squared yields 576 equals, B C squared. Taking the square root of both sides of 576 equals, B C squared yields 24 equals B C.

Question 51 51 of 269 selected Lines, Angles, And Triangles H

In triangles L M N and R S T , angles L and R each have measure 60 °, L N = 10 , and R T = 30 . Which additional piece of information is sufficient to prove that triangle L M N is similar to triangle R S T ?

  1. M N = 7 and S T = 7

  2. M N = 7 and S T = 21

  3. The measures of angles M and S are 70 ° and 60 °, respectively.

  4. The measures of angles M and T are 70 ° and 50 °, respectively.

Show Answer Correct Answer: D

Choice D is correct. Two triangles are similar if they have three pairs of congruent corresponding angles. It’s given that angles L and R each measure 60°, and so these corresponding angles are congruent. If angle M is 70°, then angle N must be 50° so that the sum of the angles in triangle L M N is 180°. If angle T is 50°, then angle S must be 70° so that the sum of the angles in triangle R S T is 180°. Therefore, if the measures of angles M and T are 70° and 50°, respectively, then corresponding angles M and S are both 70°, and corresponding angles N and T are both 50°. It follows that triangles L M N and R S T have three pairs of congruent corresponding angles, and so the triangles are similar. Therefore, the additional piece of information that is sufficient to prove that triangle L M N is similar to triangle R S T is that the measures of angles M and T are 70° and 50°, respectively.

Choice A is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the two sides given are not proportional and the angle given is not included by the given sides.

Choice B is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the angle given is not included between the proportional sides.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 52 52 of 269 selected Area And Volume M

  • One angle is a right angle.
  • The lengths of the 2 sides adjacent to the right angle are as follows:
    • 3
    • 5
  • A note indicates the figure is not drawn to scale.

The figure shows the lengths, in inches, of two sides of a right triangle. What is the area of the triangle, in square inches?

Show Answer Correct Answer: 7.5, 15/2

The correct answer is 15 2 . The area, A , of a triangle is given by the formula A=12bh, where b is the length of the base of the triangle and h is the height of the triangle. In the right triangle shown, the length of the base of the triangle is 5 inches, and the height is 3 inches. It follows that b = 5 and h = 3 . Substituting 5 for b and 3 for h in the formula A=12bh yields A=12(5)(3), which is equivalent to A=12(15), or A = 15 2 . Therefore, the area of the triangle, in square inches, is 15 2 . Note that 15/2 and 7.5 are examples of ways to enter a correct answer.

Question 53 53 of 269 selected Right Triangles And Trigonometry E

  • Angle upper C is a right angle.
  • The length of side upper A upper B is 171.
  • The length of side upper B upper C is 35.
  • A note indicates the figure is not drawn to scale.

In the right triangle shown, what is the value of sinA?

  1. 1 171

  2. 35 171

  3. 171 35

  4. 171

Show Answer Correct Answer: B

Choice B is correct. The sine of an acute angle in a right triangle is the ratio of the length of the side opposite that angle to the length of the hypotenuse. The hypotenuse of a right triangle is the side opposite the right angle. In right triangle A B C , side B C is the side opposite angle A and side A B is the hypotenuse. It's given that the length of side B C is 35 units and the length of side A B is 171 units. Therefore, the value of sinA is 35 171 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the ratio of the length of the hypotenuse to the length of the side opposite angle A rather than the ratio of the length of the side opposite angle A to the length of the hypotenuse.

Choice D is incorrect. This is the length of the hypotenuse rather than sinA.

Question 54 54 of 269 selected Right Triangles And Trigonometry E

  • One angle is a right angle.
  • The lengths of the sides adjacent to the right angle are as follows:
    • 3
    • 7
  • A note indicates the figure is not drawn to scale.

The lengths of the legs of a right triangle are shown. Which of the following is closest to the length of the triangle's hypotenuse?

  1. 3.2

  2. 5

  3. 7.6

  4. 20

Show Answer Correct Answer: C

Choice C is correct. The Pythagorean theorem states that for a right triangle, a2+b2=c2, where a and b represent the lengths of the legs of the triangle and c represents the length of its hypotenuse. In the triangle shown, the legs have lengths of 3 and 7 . Substituting 3 for a and 7 for b in the equation a2+b2=c2 yields 32+72=c2, which is equivalent to 9+49=c2, or 58 = c 2 . Taking the positive square root of both sides of this equation yields 58=c. Thus, the value of c is approximately 7.6 . Therefore, of the given choices, 7.6 is the closest to the length of the triangle's hypotenuse.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 55 55 of 269 selected Right Triangles And Trigonometry E

  • One angle is a right angle.
  • The length of the side opposite the right angle is 19.
  • The lengths of the other 2 sides are as follows:
    • b
    • 4
  • A note indicates the figure is not drawn to scale.

Which equation shows the relationship between the side lengths of the given triangle?

  1. 4b=19

  2. 4+b=19

  3. 42+b2=192

  4. 42-b2=192

Show Answer Correct Answer: C

Choice C is correct. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, a2+b2=c2, where a and b are the lengths of the legs and c is the length of the hypotenuse. For the given right triangle, the lengths of the legs are 4 and b , and the length of the hypotenuse is 19 . Substituting 4 for a and 19 for c in the equation a2+b2=c2 yields 42+b2=192. Thus, the relationship between the side lengths of the given triangle is 42+b2=192.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 56 56 of 269 selected Circles H

The graph of x 2 + x + y 2 + y = 199 2 in the xy-plane is a circle. What is the length of the circle’s radius?

Show Answer Correct Answer: 10

The correct answer is 10 . It's given that the graph of x2+x+y2+y=1992 in the xy-plane is a circle. The equation of a circle in the xy-plane can be written in the form (x-h)2+(y-k)2=r2, where the coordinates of the center of the circle are (h,k) and the length of the radius of the circle is r . The term (x-h)2 in this equation can be obtained by adding the square of half the coefficient of x to both sides of the given equation to complete the square. The coefficient of x is 1 . Half the coefficient of x is 1 2 . The square of half the coefficient of x is 1 4 . Adding 1 4 to each side of (x2+x)+(y2+y)=1992 yields (x2+x+14)+(y2+y)=1992+14, or (x+12)2+(y2+y)=1992+14. Similarly, the term (y-k)2 can be obtained by adding the square of half the coefficient of y to both sides of this equation, which yields (x+12)2+(y2+y+14)=1992+14+14, or (x+12)2+(y+12)2=1992+14+14. This equation is equivalent to (x+12)2+(y+12)2=100, or (x+12)2+(y+12)2=102. Therefore, the length of the circle's radius is 10 .

Question 57 57 of 269 selected Right Triangles And Trigonometry H

The perimeter of an isosceles right triangle is 18 + 18 2 inches. What is the length, in inches, of the hypotenuse of this triangle?

  1. 9

  2. 9 2

  3. 18

  4. 18 2

Show Answer Correct Answer: C

Choice C is correct. The perimeter of a triangle is the sum of the lengths of its sides. Since the given triangle is an isosceles right triangle, the length of each leg is the same and the length of the hypotenuse is equal to 2 times the length of a leg. Let x represent the length, in inches, of a leg of this isosceles right triangle. Therefore, the perimeter, in inches, of the triangle is x+x+x2, or 2x+x2, which is equivalent to x(2+2). It's given that the perimeter of this triangle is 18+182 inches. Thus, x(2+2)=18+182. Dividing both sides of this equation by 2+2 yields x=18+1822+2. Multiplying the right-hand side of this equation by 2-22-2 yields x=36+362-182-362, or x=92. It follows that the length, in inches, of a leg of this isosceles right triangle is 92. Therefore, the length, in inches, of the hypotenuse of this isosceles right triangle is (92)(2), or 18.  

Choice A is incorrect. If this were the length of the hypotenuse, the perimeter would be 9+92 inches.

Choice B is incorrect. This is the length, in inches, of a leg of this triangle, not the hypotenuse.

Choice D is incorrect. If this were the length of the hypotenuse, the perimeter would be 36+182 inches.

Question 58 58 of 269 selected Area And Volume M

A cube has an edge length of 41 inches. What is the volume, in cubic inches, of the cube?

  1. 164

  2. 1,681

  3. 10,086

  4. 68,921

Show Answer Correct Answer: D

Choice D is correct. The volume, V , of a cube can be found using the formula V=s3, where s is the edge length of the cube. It's given that a cube has an edge length of 41 inches. Substituting 41 inches for s in this equation yields V=413 cubic inches, or V = 68,921 cubic inches. Therefore, the volume of the cube is 68,921 cubic inches.

Choice A is incorrect. This is the perimeter, in inches, of the cube.

Choice B is incorrect. This is the area, in square inches, of a face of the cube.

Choice C is incorrect. This is the surface area, in square inches, of the cube.

Question 59 59 of 269 selected Area And Volume M

The length of the edge of the base of a right square prism is 6 units. The volume of the prism is 2,880 cubic units. What is the height, in units, of the prism?

  1. 4 30

  2. 36

  3. 24 5

  4. 80

Show Answer Correct Answer: D

Choice D is correct. The volume, V , of a right square prism is given by the formula V=s2h, where s represents the length of the edge of the base and h represents the height of the prism. It’s given that the volume of a right square prism is 2,880 cubic units and the length of the edge of the base is 6 units. Substituting 2,880 for V and 6 for s in the formula V=s2h yields 2,880=(62)h, or 2,880 = 36 h . Dividing both sides of this equation by 36 yields 80 = h . Therefore, the height, in units, of the prism is 80 .

Choice A is incorrect. This is the height, in units, of a right square prism where the length of the edge of the base is 6 units and the volume of the prism is 14430, not 2,880 , units.

Choice B is incorrect. This is the area, in square units, of the base, not the height, in units, of the prism.

Choice C is incorrect. This is the height, in units, of a right square prism where the length of the edge of the base is 6 units and the volume of the prism is 8645, not 2,880 , units.

Question 60 60 of 269 selected Area And Volume M

A cylinder has a diameter of 8 inches and a height of 12 inches. What is the volume, in cubic inches, of the cylinder?

  1. 16 π

  2. 96 π

  3. 192 π

  4. 768 π

Show Answer Correct Answer: C

Choice C is correct. The base of a cylinder is a circle with a diameter equal to the diameter of the cylinder. The volume, V , of a cylinder can be found by multiplying the area of the circular base, A , by the height of the cylinder, h , or V=Ah. The area of a circle can be found using the formula A=πr2, where r is the radius of the circle. It’s given that the diameter of the cylinder is 8 inches. Thus, the radius of this circle is 4 inches. Therefore, the area of the circular base of the cylinder is A=π(4)2, or 16π square inches. It’s given that the height h of the cylinder is 12 inches. Substituting 16π for A and 12 for h in the formula V=Ah gives V=16π(12), or 192π cubic inches.

Choice A is incorrect. This is the area of the circular base of the cylinder.

Choice B is incorrect and may result from using 8 , instead of 16 , as the value of r2 in the formula for the area of a circle.

Choice D is incorrect and may result from using 8 , instead of 4 , for the radius of the circular base.

Question 61 61 of 269 selected Right Triangles And Trigonometry H

  • The length of side upper A upper B is 54.
  • The measure of angle upper B is 30°.
  • Angle upper C is a right angle.
  • A note indicates the figure is not drawn to scale.

Right triangle A B C is shown. What is the value of tanA?

  1. 354

  2. 13

  3. 3

  4. 273

Show Answer Correct Answer: C

Choice C is correct. In the triangle shown, the measure of angle B is 30° and angle C is a right angle, which means that it has a measure of 90°. Since the sum of the angles in a triangle is equal to 180°, the measure of angle A is equal to 180°-(30+90)°, or 60°. In a right triangle whose acute angles have measures 30° and 60°, the lengths of the legs can be represented by the expressions x , x3, and 2 x , where x is the length of the leg opposite the angle with measure 30°, x3 is the length of the leg opposite the angle with measure 60°, and 2 x is the length of the hypotenuse. In the triangle shown, the hypotenuse has a length of 54 . It follows that 2x=54, or x=27. Therefore, the length of the leg opposite angle B is 27 and the length of the leg opposite angle A is 273. The tangent of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. The length of the leg opposite angle A is 273 and the length of the leg adjacent to angle A is 27 . Therefore, the value of tanA is 27327, or 3.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of 1tanA, not the value of tanA.

Choice D is incorrect. This is the length of the leg opposite angle A , not the value of tanA.

Question 62 62 of 269 selected Area And Volume M

Square A has side lengths that are 246 times the side lengths of square B. The area of square A is k times the area of square B. What is the value of k ?

  1. 60,516

  2. 492

  3. 246

  4. 123

Show Answer Correct Answer: A

Choice A is correct. The area of a square is s2, where s is the side length of the square. Therefore, the area of square B is b2, where b is the side length of square B. It’s given that square A has side lengths that are 246 times the side lengths of square B. Therefore, the side length of square A can be represented by the expression 246b. It follows that the area of square A is (246b)2, or 60,516b2. It’s given that the area of square A is k times the area of square B, so 60,516b2=kb2. Therefore, the value of k is 60,516.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 63 63 of 269 selected Area And Volume M

A manufacturing company produces two sizes of cylindrical containers that each have a height of 50 centimeters. The radius of container A is 16 centimeters, and the radius of container B is 25% longer than the radius of container A. What is the volume, in cubic centimeters, of container B?

  1. 16,000 pi

  2. 20,000 pi

  3. 25,000 pi

  4. 31,250 pi

Show Answer Correct Answer: B

Choice B is correct. If the radius of container A is 16 centimeters and the radius of container B is 25% longer than the radius of container A, then the radius of container B is 16 plus, 0 point 2 5 times 16, equals 20 centimeters. The volume of a cylinder is pi, times r squared, times h, where r is the radius of the cylinder and h is its height. Substituting r equals 20 and h equals 50 into pi, times r squared, times h yields that the volume of cylinder B is pi times open parenthesis, 20, close parenthesis, squared, times 50, equals 20,000 pi cubic centimeters.

Choice A is incorrect and may result from multiplying the radius of cylinder B by the radius of cylinder A rather than squaring the radius of cylinder B. Choice C is incorrect and may result from multiplying the radius of cylinder B by 25 rather than squaring it. Choice D is incorrect and may result from taking the radius of cylinder B to be 25 centimeters rather than 20 centimeters.

 

Question 64 64 of 269 selected Circles H

Point F lies on a unit circle in the xy-plane and has coordinates (1,0). Point G is the center of the circle and has coordinates (0,0). Point H also lies on the circle and has coordinates (-1,y), where y is a constant. Which of the following could be the positive measure of angle F G H , in radians?

  1. 27π2

  2. 29π2

  3. 24π

  4. 25π

Show Answer Correct Answer: D

Choice D is correct. It's given that the circle is a unit circle, which means the circle has a radius of 1. It's also given that point G is the center of the circle and has coordinates (0,0) and that point H lies on the circle and has coordinates (-1,y). Since the radius of the circle is 1, the value of y must be 0, as all other points with an x-coordinate of -1 are a distance greater than 1 away from point G. Since F and H are points on the unit circle centered at G, let side FG be the starting side of the angle and side GH be the terminal side of the angle. Since side FG is on the positive x-axis and side GH is on the negative x-axis, side FG is half of a rotation around the unit circle, or π radians, away from side GH. Therefore, the positive measure of angle FGH, in radians, is equal to π plus an integer multiple of 2π. In other words, the positive measure of angle FGH, in radians, is an odd integer multiple of π. Of the given choices, only 25π is an odd integer multiple of π.

Choice A is incorrect. This could be the positive measure of an angle where the starting side is FG and the terminal side contains the point (0,-1), not (-1,0).

Choice B is incorrect. This could be the positive measure of an angle where the starting side is FG and the terminal side contains the point (0,1), not (-1,0).

Choice C is incorrect. This could be the positive measure of an angle where the starting side is FG and the terminal side contains the point (1,0), not (-1,0).

Question 65 65 of 269 selected Circles H

(x+4)2+(y-19)2=121

The graph of the given equation is a circle in the xy-plane. The point (a,b) lies on the circle. Which of the following is a possible value for a ?

  1. -16

  2. -14

  3. 11

  4. 19

Show Answer Correct Answer: B

Choice B is correct. An equation of the form (x-h)2+(y-k)2=r2, where h , k , and r are constants, represents a circle in the xy-plane with center (h,k) and radius r . Therefore, the circle represented by the given equation has center (-4,19) and radius 11 . Since the center of the circle has an x-coordinate of -4 and the radius of the circle is 11 , the least possible x-coordinate for any point on the circle is -4-11, or -15 . Similarly, the greatest possible x-coordinate for any point on the circle is -4+11, or 7 . Therefore, if the point (a,b) lies on the circle, it must be true that -15a7. Of the given choices, only -14 satisfies this inequality.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 66 66 of 269 selected Area And Volume H

Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 58 square centimeters (cm2), and the surface area of right rectangular prism Y is 1,450 cm2. The volume of right rectangular prism Y is 1,250 cubic centimeters (cm3). What is the sum of the volumes, in cm3, of right rectangular prism X and right rectangular prism Y?

Show Answer Correct Answer: 1260

The correct answer is 1,260 . Since it's given that prisms X and Y are similar, all the linear measurements of prism Y are k times the respective linear measurements of prism X, where k is a positive constant. Therefore, the surface area of prism Y is k 2 times the surface area of prism X and the volume of prism Y is k 3 times the volume of prism X. It's given that the surface area of prism Y is 1,450 cm2, and the surface area of prism X is 58 cm2, which implies that 1,450 = 58 k 2 . Dividing both sides of this equation by 58 yields 1,45058=k2, or k 2 = 25 . Since k is a positive constant, k = 5 . It's given that the volume of prism Y is 1,250 cm3. Therefore, the volume of prism X is equal to 1,250 k 3  cm3, which is equivalent to 1,25053 cm3, or 10 cm3. Thus, the sum of the volumes, in cm3, of the two prisms is 1,250+10, or 1,260 .

Question 67 67 of 269 selected Circles H

x squared, plus 20 x, plus y squared, plus 16 y, equals negative 20

The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?

  1. negative 20 comma negative 16

  2. negative 10 comma negative 8

  3. 10 comma 8

  4. 20 comma 16

Show Answer Correct Answer: B

Choice B is correct. The standard equation of a circle in the xy-plane is of the form open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared, where the ordered pair h comma k are the coordinates of the center of the circle and r is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the first two terms, x squared, plus 20 x, needs a 100 to complete the square, and the sum of the second two terms, y squared, plus 16 y, needs a 64 to complete the square. Adding 100 and 64 to both sides of the given equation yields open parenthesis, x squared, plus 20 x, plus 100, close parenthesis, plus, open parenthesis, y squared, plus 16 y, plus 64, close parenthesis, equals negative 20, plus 100, plus 64, which is equivalent to open parenthesis, x plus 10, close parenthesis, squared, plus, open parenthesis, y plus 8, close parenthesis, squared, equals 144. Therefore, the coordinates of the center of the circle are negative 10 comma negative 8.

Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete the squares or when identifying the coordinates of the center.

 

Question 68 68 of 269 selected Area And Volume E

A rectangle has an area of 63 square meters and a length of 9 meters. What is the width, in meters, of the rectangle?

  1. 7

  2. 54

  3. 81

  4. 567

Show Answer Correct Answer: A

Choice A is correct. The area A , in square meters, of a rectangle is the product of its length l, in meters, and its width w , in meters; thus, A=lw. It's given that a rectangle has an area of 63 square meters and a length of 9 meters. Substituting 63 for A and 9 for l in the equation A=lw yields 63 = 9 w . Dividing both sides of this equation by 9 yields 7 = w . Therefore, the width, in meters, of the rectangle is 7 .

Choice B is incorrect. This is the difference between the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle.

Choice C is incorrect. This is the square of the length, in meters, not the width, in meters, of the rectangle.

Choice D is incorrect. This is the product of the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle.

Question 69 69 of 269 selected Area And Volume M

A sphere has a radius of 17 5 feet. What is the volume, in cubic feet, of the sphere?

  1. 5 π 17

  2. 68 π 15

  3. 32 π 5

  4. 19,652 π 375

Show Answer Correct Answer: D

Choice D is correct. The volume, V , of a sphere can be found using the formula V=43πr3, where r is the radius of the sphere. It’s given that the sphere has a radius of 17 5 feet. Substituting 17 5 for r in the formula V=43πr3 yields V=43π(175)3, which is equivalent to V=43π(4,913125), or V=19,652π375. Therefore, the volume, in cubic feet, of the sphere is 19,652π375.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the volume, in cubic feet, of a sphere with a radius of 1753 feet.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 70 70 of 269 selected Right Triangles And Trigonometry H

  • One angle is a right angle.
  • The measure of a second angle is x°.
  • The length of the hypotenuse is 23.
  • The length of the leg opposite the angle with measure x° is 16.
  • Note: Figure not drawn to scale.

In the triangle shown, what is the value of sinx°?

Show Answer Correct Answer: .6956, .6957, 16/23

The correct answer is 16 23 . In a right triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the length of the side opposite the angle with measure x° is 16 units and the length of the hypotenuse is 23 units. Therefore, the value of sinx° is 16 23 . Note that 16/23, .6956, .6957, 0.695, and 0.696 are examples of ways to enter a correct answer.

Question 71 71 of 269 selected Circles H
The figure presents a circle with center O. Point A, lies on the upper left part of the circle. Point B lies on the upper right part of the circle. Point C is indicated at the bottom of the circle, directly below the center O. Line segment O A, line segment O B, and horizontal line segment A, B are drawn forming triangle O A, B. Vertical line segment O C is drawn.

Point O is the center of the circle above, and the measure of angle O A, B is 30 degrees. If the length of line segment O C is 18, what is the length of arc A, B?

  1. 9 pi

  2. 12 pi

  3. 15 pi

  4. 18 pi

Show Answer Correct Answer: B

Choice B is correct. Because segments OA and OB are radii of the circle centered at point O, these segments have equal lengths. Therefore, triangle AOB is an isosceles triangle, where angles OAB and OBA are congruent base angles of the triangle. It’s given that angle OAB measures 30 degrees. Therefore, angle OBA also measures 30 degrees. Let x degrees represent the measure of angle AOB. Since the sum of the measures of the three angles of any triangle is 180 degrees, it follows that 30 degrees plus 30 degrees, plus x degrees, equals 180 degrees, or 60 degrees plus x degrees, equals 180 degrees. Subtracting 60 degrees from both sides of this equation yields x degrees equals 120 degrees, or the fraction 2 pi over 3 radians. Therefore, the measure of angle AOB, and thus the measure of arc A, B, is the fraction 2 pi over 3 radians. Since the line segment O C is a radius of the given circle and its length is 18, the length of the radius of the circle is 18. Therefore, the length of arc A, B can be calculated as the fraction 2 pi over 3, end fraction, times 18, or 12 pi.

Choices A, C, and D are incorrect and may result from conceptual or computational errors.

 

Question 72 72 of 269 selected Right Triangles And Trigonometry M

In ABCB is a right angle and the length of BC¯ is 136 millimeters. If cosA=35, what is the length, in millimeters, of AB¯?

  1. 34

  2. 102

  3. 136

  4. 170

Show Answer Correct Answer: B

Choice B is correct. It's given that in ABCB is a right angle. Therefore, ABC is a right triangle, and AC¯ is the hypotenuse of the triangle. It's also given that cosA=35. Since the cosine of an acute angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, the ratio of the length of AB¯ to the length of AC¯ is 3 to 5. It follows that the length of AB¯ can be represented as 3a and the length of AC¯ can be represented as 5a, where a is a constant. The Pythagorean theorem states that the sum of the squares of the length of the legs of a right triangle is equal to the square of the length of its hypotenuse, so it follows that AB2+BC2=AC2. Substituting 3a for AB and 5a for AC in this equation yields (3a)2+BC2=(5a)2, or 9a2+BC2=25a2. Subtracting 9a2 from both sides of this equation yields BC2=16a2, or BC=4a. It follows that the ratio of the length of AB¯ to the length of BC¯ is 3 to 4. Let x represent the length, in millimeters, of AB¯. It's given that the length of BC¯ is 136 millimeters. Since the ratio of the length of AB¯ to the length of BC¯ is 3 to 4x136=34. Multiplying both sides of this equation by 136 yields x=3(136)4, or x=102. Therefore, the length of AB¯ is 102 millimeters.

Choice A is incorrect. This is the scale factor by which the 3 to 4 to 5 ratio is multiplied that results in the side lengths of ABC.

Choice C is incorrect. This is the length of BC¯, not the length of AB¯.

Choice D is incorrect. This is the length of AC¯, not the length of AB¯.

Question 73 73 of 269 selected Circles H

A circle in the xy-plane has a diameter with endpoints (2,4) and (2,14). An equation of this circle is ( x - 2 ) 2 + ( y - 9 ) 2 = r 2 , where r is a positive constant. What is the value of r ?

Show Answer Correct Answer: 5

The correct answer is 5 . The standard form of an equation of a circle in the xy-plane is (x-h)2+(y-k)2=r2, where h , k , and r are constants, the coordinates of the center of the circle are (h,k), and the length of the radius of the circle is r . It′s given that an equation of the circle is (x-2)2+(y-9)2=r2. Therefore, the center of this circle is (2,9). It’s given that the endpoints of a diameter of the circle are (2,4) and (2,14). The length of the radius is the distance from the center of the circle to an endpoint of a diameter of the circle, which can be found using the distance formula, (x1-x2)2+(y1-y2)2. Substituting the center of the circle (2,9) and one endpoint of the diameter (2,4) in this formula gives a distance of (2-2)2+(9-4)2, or 02+52, which is equivalent to 5 . Since the distance from the center of the circle to an endpoint of a diameter is 5 , the value of r is 5 .

Question 74 74 of 269 selected Lines, Angles, And Triangles M

Each side of equilateral triangle S is multiplied by a scale factor of k to create equilateral triangle T. The length of each side of triangle T is greater than the length of each side of triangle S. Which of the following could be the value of k ?

  1. 29 28

  2. 1

  3. 28 29

  4. 0

Show Answer Correct Answer: A

Choice A is correct. It's given that each side of equilateral triangle S is multiplied by a scale factor of k to create equilateral triangle T. Since the length of each side of triangle T is greater than the length of each side of triangle S, the scale factor of k must be greater than 1 . Of the given choices, only 29 28 is greater than 1 .

Choice B is incorrect. If each side of equilateral triangle S is multiplied by a scale factor of 1 , the length of each side of triangle T would be equal to the length of each side of triangle S.

Choice C is incorrect. If each side of equilateral triangle S is multiplied by a scale factor of 28 29 , the length of each side of triangle T would be less than the length of each side of triangle S.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 75 75 of 269 selected Area And Volume M

A right circular cylinder has a height of 8 meters (m) and a base with a radius of 12 m. What is the volume, in m3, of the cylinder?

  1. 8 π

  2. 20 π

  3. 768 π

  4. 1,152 π

Show Answer Correct Answer: D

Choice D is correct. The volume, V, of a right circular cylinder is given by V=πr2h, where r is the radius of the circular base and h is the height of the cylinder. It’s given that the cylinder has a height of 8 meters and a base with a radius of 12 meters. Substituting 12 for r and 8 for h in V=πr2h yields V=π(12)2(8), or V=1,152π. Therefore, the volume, in m3, of the cylinder is 1,152π.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the volume, in m3, of a cylinder with a radius of 8 meters and a height of 12 meters.

Question 76 76 of 269 selected Area And Volume H

Rectangle ABCD is similar to rectangle EFGH. The area of rectangle ABCD is 648 square inches, and the area of rectangle EFGH is 72 square inches. The length of the longest side of rectangle ABCD is 36 inches. What is the length, in inches, of the longest side of rectangle EFGH?

  1. 4

  2. 9

  3. 12

  4. 36

Show Answer Correct Answer: C

Choice C is correct. It's given that rectangle ABCD is similar to rectangle EFGH. Therefore, if the length of each side of rectangle ABCD is k times the length of the corresponding side of rectangle EFGH, then the area of rectangle ABCD is k2 times the area of rectangle EFGH. It’s given that the area of rectangle ABCD is 648 square inches and the area of rectangle EFGH is 72 square inches. It follows that k2=64872, or k2=9. Taking the square root of each side of this equation yields k=9, or k=3. It follows that the length of each side of rectangle ABCD is 3 times the length of the corresponding side of rectangle EFGH. It’s given that the length of the longest side of rectangle ABCD is 36 inches. Therefore, 36 inches is 3 times the length of the longest side of rectangle EFGH, and the longest side of rectangle EFGH is equal to 363, or 12, inches.

Choice A is incorrect. This is the length, in inches, of the longest side of a rectangle with side lengths that are 19 the corresponding side lengths of rectangle ABCD, rather than a rectangle with an area that is 19 the area of rectangle ABCD.

Choice B is incorrect. This is the factor by which the area of rectangle ABCD is larger than the area of rectangle EFGH, not the length, in inches, of the longest side of rectangle EFGH.

Choice D is incorrect. This is the length, in inches, of the longest side of rectangle ABCD, not rectangle EFGH.

Question 77 77 of 269 selected Right Triangles And Trigonometry H
The figure presents a trapezoid, which consists of three congruent equilateral triangles. Two of the triangles are shaded, and each of those triangles shares a side with the unshaded triangle.

A graphic designer is creating a logo for a company. The logo is shown in the figure above. The logo is in the shape of a trapezoid and consists of three congruent equilateral triangles. If the perimeter of the logo is 20 centimeters, what is the combined area of the shaded regions, in square centimeters, of the logo?

  1. 2 times the square root of 3

  2. 4 times the square root of 3

  3. 8 times the square root of 3

  4. 16

Show Answer Correct Answer: C

Choice C is correct. It’s given that the logo is in the shape of a trapezoid that consists of three congruent equilateral triangles, and that the perimeter of the trapezoid is 20 centimeters (cm). Since the perimeter of the trapezoid is the sum of the lengths of 5 of the sides of the triangles, the length of each side of an equilateral triangle is the fraction 20 over 5 equals 4 centimeters. Dividing up one equilateral triangle into two right triangles yields a pair of congruent 30°-60°-90° triangles. The shorter leg of each right triangle is half the length of the side of an equilateral triangle, or 2 cm. Using the Pythagorean Theorem, a, squared, plus b squared, equals c squared, the height of the equilateral triangle can be found. Substituting a, equals 2 and c equals 4 and solving for b yields the square root of, 4 squared, minus 2 squared, end root, equals the square root of 12, which equals, 2 times the square root of 3 centimeters cm. The area of one equilateral triangle is one half b h, where b equals 2 and h equals, 2 times the square root of 3. Therefore, the area of one equilateral triangle is one half times 4, times, open parenthesis, 2 times the square root of 3, close parenthesis, equals, 4 times the square root of 3 centimeters squared. The shaded area consists of two such triangles, so its area is 2 times 4, times the square root of 3, equals, 8 times the square root of 3 centimeters squared.

Alternate approach: The area of a trapezoid can be found by evaluating the expression one half times, open parenthesis, b sub 1 plus b sub 2, close parenthesis, times h, where b sub 1is the length of one base, b sub 2 is the length of the other base, and h is the height of the trapezoid. Substituting b sub 1 equals 8, b sub 2 equals 4, and h equals, 2 times the square root of 3 yields the expression one half times, open parenthesis, 8 plus 4, close parenthesis, times, open parenthesis, 2 times the square root of 3, close parenthesis, or one half times 12, times, open parenthesis, 2 times the square root of 3, close parenthesis, which gives an area of 12 times the square root of 3 centimeters squared for the trapezoid. Since two-thirds of the trapezoid is shaded, the area of the shaded region is two thirds times, 12 times the square root of 3, equals, 8 times the square root of 3.

Choice A is incorrect. This is the height of the trapezoid. Choice B is incorrect. This is the area of one of the equilateral triangles, not two. Choice D is incorrect and may result from using a height of 4 for each triangle rather than the height of 2 times the square root of 3.

 

Question 78 78 of 269 selected Circles H

Circle A in the xy-plane has the equation ( x + 5 ) 2 + ( y - 5 ) 2 = 4 . Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A. The equation defining circle B in the xy-plane is ( x + 5 ) 2 + ( y - 5 ) 2 = k , where k is a constant. What is the value of k ?

Show Answer Correct Answer: 16

The correct answer is 16 . An equation of a circle in the xy-plane can be written as (x-t)2+(y-u)2=r2, where the center of the circle is (t,u) , the radius of the circle is r , and where t , u , and r are constants. It’s given that the equation of circle A is (x+5)2+(y-5)2=4, which is equivalent to (x+5)2+(y-5)2=22. Therefore, the center of circle A is (-5,5) and the radius of circle A is 2 . It’s given that circle B has the same center as circle A and that the radius of circle B is two times the radius of circle A. Therefore, the center of circle B is (-5,5) and the radius of circle B is 2(2), or 4 . Substituting -5 for t , 5 for u , and 4 for r into the equation (x-t)2+(y-u)2=r2  yields (x+5)2+(y-5)2=42, which is equivalent to (x+5)2+(y-5)2=16. It follows that the equation of circle B in the xy-plane is (x+5)2+(y-5)2=16. Therefore, the value of k is 16 .

Question 79 79 of 269 selected Circles H

What is the diameter of the circle in the xy-plane with equation ( x - 5 ) 2 + ( y - 3 ) 2 = 16 ?

  1. 4

  2. 8

  3. 16

  4. 32

Show Answer Correct Answer: B

Choice B is correct. The standard form of an equation of a circle in the xy-plane is (x-h)2+(y-k)2=r2, where the coordinates of the center of the circle are (h,k) and the length of the radius of the circle is r . For the circle in the xy-plane with equation (x-5)2+(y-3)2=16, it follows that r2=16. Taking the square root of both sides of this equation yields r = 4 or r = -4 . Because r represents the length of the radius of the circle and this length must be positive, r = 4 . Therefore, the radius of the circle is 4 . The diameter of a circle is twice the length of the radius of the circle. Thus, 2(4) yields 8 . Therefore, the diameter of the circle is 8 .

Choice A is incorrect. This is the radius of the circle. 

Choice C is incorrect. This is the square of the radius of the circle. 

Choice D is incorrect and may result from conceptual or calculation errors.

Question 80 80 of 269 selected Lines, Angles, And Triangles M
The figure presents two right triangles, A, B C and D E F. In triangle A, B C, side A, C is horizontal and side B C is drawn such that it is perpendicular to side A, C and point B, is above point C.  Angle A, is labeled 32 degrees and the right angle symbol is indicated at angle C. In triangle D E F, side D F is horizontal, side E F is drawn such that it is perpendicular to side D F, and point E, is above point F. Angle D is labeled 58 degrees and the right angle symbol is indicated at angle F.

Triangles ABC and DEF are shown above. Which of the following is equal to the ratio B C over A, B?

  1. D E over D F

  2. D F over D E

  3. D F over E F

  4. E F over D E

Show Answer Correct Answer: B

Choice B is correct. In right triangle ABC, the measure of angle B must be 58° because the sum of the measure of angle A, which is 32°, and the measure of angle B is 90°. Angle D in the right triangle DEF has measure 58°. Hence, triangles ABC and DEF are similar (by angle-angle similarity). Since side B C is the side opposite to the angle with measure 32° and AB is the hypotenuse in right triangle ABC, the ratio the length of side B C over the length of side A, B is equal to the length of side D F over the length of side D E.

Alternate approach: The trigonometric ratios can be used to answer this question. In right triangle ABC, the ratio the length of side B C over the length of side A, B equals, the sine of 32 degrees. The angle E in triangle DEF has measure 32° because the measure of angle D, plus the measure of angle E, equals 90 degrees. In triangle DEF, the ratio the length of side D F over the length of side D E, equals the sine of 32 degrees. Therefore, the length of side D F over the length of side D E, equals, the length of side B C over the length of side A, B.

Choice A is incorrect because the length of side D E over the length of side D F is the reciprocal of the ratio the length of side B C over the length of side A, B. Choice C is incorrect because the length of side D F over the length of side D E, equals the length of side B C over the length of side A, C, not the length of side B C over the length of side A, B. Choice D is incorrect because the length of side E F over the length of side D E, equals, the length of side A, C over the length of side A, B, not the length of side B C over the length of side A, B.

 

Question 81 81 of 269 selected Right Triangles And Trigonometry E

  • One angle is a right angle.
  • The measure of a second angle is x°.
  • The length of the leg opposite the angle with measure x° is 26.
  • The length of the leg adjacent to the angle with measure x° is 7.
  • A note indicates the figure is not drawn to scale.

In the triangle shown, what is the value of tan x°?

  1. 126

  2. 1926

  3. 267

  4. 337

Show Answer Correct Answer: C

Choice C is correct. The tangent of an acute angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the shorter side adjacent to the angle. In the triangle shown, the length of the side opposite the angle with measure x° is 26 units and the length of the side adjacent to the angle with measure x° is 7 units. Therefore, the value of tanx° is 267.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 82 82 of 269 selected Lines, Angles, And Triangles M

  • Line segment upper B upper D is drawn from side upper C upper A to side upper C upper E to form triangle upper C upper B upper D.
  • Angle upper C upper D upper B is a right angle.
  • Angle upper C upper E upper A is a right angle.
  • A note indicates the figure is not drawn to scale.


In the figure shown, triangle CAE is similar to triangle CBD. The measure of angle CBD is 57°, and AE=26(BD). What is the measure of angle CAE?

  1. (26·57)°

  2. (26+57)°

  3. 57°

  4. 26°

Show Answer Correct Answer: C

Choice C is correct. It's given that triangle CAE is similar to triangle CBD. Corresponding angles in similar triangles have equal measure. Angle BCD and angle ACE represent the same angle. It follows that angle BCD and angle ACE have equal measure and are corresponding angles. It's given in the figure that angle BDC and angle AEC are right angles and therefore have equal measure. It follows that angle BDC and angle AEC are corresponding angles. Therefore, angle CBD and angle CAE are corresponding angles and have equal measure. It's given that the measure of angle CBD is 57°, so the measure of angle CAE is 57°.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 83 83 of 269 selected Area And Volume H

  • From left to right, the 3 points have the following coordinates:
    • (2 comma 6)
    • (6 comma 10)
    • (6 comma 2)

The three points shown define a circle. The circumference of this circle is kπ, where k is a constant. What is the value of k ?

Show Answer Correct Answer: 8

The correct answer is 8. It's given that the three points shown define a circle, so the center of that circle is an equal distance from each of the three points. The point (6,6) is halfway between the points (6,2) and (6,10), and is a distance of 4 units from each of those two points. The point (6,6) is also a distance of 4 units from (2,6). Because the point (6,6) is the same distance from all three points shown, it must be the center of the circle. Since that distance is 4, it follows that the radius of the circle is 4. The circumference of a circle with radius r is equal to 2πr. It follows that the circumference of the given circle is 2π(4), or 8π. It's given that the circumference of the circle is kπ. Therefore, the value of k is 8.

Question 84 84 of 269 selected Area And Volume H

The floor of a ballroom has an area of 600 square meters. An architect creates a scale model of the floor of the ballroom, where the length of each side of the model is 1 10 times the length of the corresponding side of the actual floor of the ballroom. What is the area, in square meters, of the scale model?

  1. 6

  2. 10

  3. 60

  4. 150

Show Answer Correct Answer: A

Choice A is correct. It’s given that the length of each side of a scale model is 1 10 times the length of the corresponding side of the actual floor of a ballroom. Therefore, the area of the scale model is (110)2, or 1 100 , times the area of the actual floor of the ballroom. It’s given that the area of the floor of the ballroom is 600 square meters. Therefore, the area, in square meters, of the scale model is (1100)(600), or 6 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 85 85 of 269 selected Area And Volume H

A cube has a volume of 474,552 cubic units. What is the surface area, in square units, of the cube?

Show Answer Correct Answer: 36504

The correct answer is 36,504 . The volume of a cube can be found using the formula V = s 3 , where s represents the edge length of a cube. It’s given that this cube has a volume of 474,552 cubic units. Substituting 474,552 for V in V=s3 yields 474,552 = s 3 . Taking the cube root of both sides of this equation yields 78=s. Thus, the edge length of the cube is 78 units. Since each face of a cube is a square, it follows that each face has an edge length of 78 units. The area of a square can be found using the formula A = s 2 . Substituting 78 for s in this formula yields A=782, or A = 6,084 . Therefore, the area of one face of this cube is 6,084 square units. Since a cube has 6 faces, the surface area, in square units, of this cube is 6(6,084), or 36,504 .

Question 86 86 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled script l, j, and k.
  • Line script l intersects both line j and line k.
  • At the intersection of line script l and line j, 2 angles are labeled clockwise from top left as follows:
    • Top right: w°
    • Bottom left: x°
  • At the intersection of line script l and line k, 3 angles are labeled clockwise from top left as follows:
    • Top right: 40°
    • Bottom right: z°
    • Bottom left: y°
  • A note indicates the figure is not drawn to scale.

In the figure shown, line l intersects lines j and k . Which additional piece of information is sufficient to prove that lines j and k are parallel?

  1. w = 40

  2. x = 140

  3. y = 40

  4. z = 140

Show Answer Correct Answer: A

Choice A is correct. In the figure shown, lines j and k are parallel if and only if a pair of corresponding angles are congruent. It's given that one angle has a measure of 40° and that the corresponding angle has a measure of w°. Therefore, w=40 is sufficient to prove that lines j and k are parallel.

Choice B is incorrect. The angle measuring x° and the angle measuring 40° are alternate interior angles. Thus, if lines j and k are parallel, x is equal to 40, not 140.

Choice C is incorrect. The angle measuring y° and the angle measuring 40° are vertical angles. Thus, y=40, whether lines j and k are parallel or not.

Choice D is incorrect. The angle measuring z° is supplementary to the angle measuring 40°. Thus, z=180-40, or z=140, whether lines j and k are parallel or not.

Question 87 87 of 269 selected Lines, Angles, And Triangles H

In right triangle A B C , angle C is the right angle and B C = 162 . Point D on side A B is connected by a line segment with point E on side A C such that line segment D E is parallel to side B C and C E = 2 A E . What is the length of line segment D E ?

Show Answer Correct Answer: 54

The correct answer is 54 . It’s given that in triangle A B C , point D on side A B is connected by a line segment with point E on side A C such that line segment D E is parallel to side B C . It follows that parallel segments D E and B C are intersected by sides A B and A C . If two parallel segments are intersected by a third segment, corresponding angles are congruent. Thus, corresponding angles C and AED are congruent and corresponding angles B and A D E are congruent. Since triangle A D E has two angles that are each congruent to an angle in triangle A B C , triangle A D E is similar to triangle A B C by the angle-angle similarity postulate, where side D E corresponds to side B C , and side A E corresponds to side A C . Since the lengths of corresponding sides in similar triangles are proportional, it follows that DEBC=AEAC. Since point E lies on side A C , A E + C E = A C . It's given that C E = 2 A E . Substituting 2 A E for C E in the equation A E + C E = A C yields AE+2AE=AC, or 3 A E = A C . It’s given that B C = 162 . Substituting 162 for B C and 3 A E for A C in the equation DEBC=AEAC yields DE162=AE3AE, or DE162=13. Multiplying both sides of this equation by 162 yields D E = 54 . Thus, the length of line segment D E is 54 .

Question 88 88 of 269 selected Area And Volume E

The perimeter of triangle A B C is 17 inches, the length of side AB is 4 inches, and the length of side AC is 7 inches. What is the length, in inches, of side BC?

  1. 4

  2. 6

  3. 7

  4. 11

Show Answer Correct Answer: B

Choice B is correct. The perimeter of a triangle is the sum of the lengths of all three sides of the triangle. It’s given that the lengths of side A B and side A C are 4 inches and 7 inches, respectively. Let x represent the length, in inches, of side B C . The sum of the lengths, in inches, of all three sides of triangle A B C can be represented by the expression 4+7+x. Since it’s given that the perimeter of triangle A B C is 17 inches, it follows that 17=4+7+x, or 17=11+x. Subtracting 11 from both sides of this equation yields 6 = x . Therefore, the length, in inches, of side B C is 6 .

Choice A is incorrect. This is the length, in inches, of side A B .

Choice C is incorrect. This is the length, in inches, of side A C .

Choice D is incorrect. This is the sum of the lengths, in inches, of sides A B and A C .

Question 89 89 of 269 selected Area And Volume M

The table gives the perimeters of similar triangles T U V and X Y Z , where TU corresponds to XY. The length of TU is 18 .

 

  Perimeter
Triangle T U V 37
Triangle X Y Z 333

 

What is the length of XY?

  1. 2

  2. 18

  3. 55

  4. 162

Show Answer Correct Answer: D

Choice D is correct. It's given that triangle XYZ is similar to triangle TUV. Therefore, each side of triangle XYZ is k times its corresponding side of triangle TUV, where k is a constant. It follows that the perimeter of triangle XYZ is k times the perimeter of triangle TUV. It's also given that TU¯ corresponds to XY¯ and the length of TU¯ is 18 . Let x represent the length of XY¯. It follows that x=18k. The table shows that the perimeters of triangles TUV and XYZ are 37 and 333 , respectively. It follows that 333=37k, or 9=k. Substituting 9 for k in the equation x=18k yields x=(18)(9), or x=162. Therefore, the length of XY¯ is 162 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the length of TU¯, not the length of XY¯.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 90 90 of 269 selected Right Triangles And Trigonometry H

The perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is k3 centimeters, where k is a constant. What is the value of k ?

Show Answer Correct Answer: 104

The correct answer is 104 . An equilateral triangle is a triangle in which all three sides have the same length and all three angles have a measure of 60°. The height of the triangle, k3, is the length of the altitude from one vertex. The altitude divides the equilateral triangle into two congruent 30-60-90 right triangles, where the altitude is the side across from the 60° angle in each 30-60-90 right triangle. Since the altitude has a length of k3, it follows from the properties of 30-60-90 right triangles that the side across from each 30° angle has a length of k and each hypotenuse has a length of 2 k . In this case, the hypotenuse of each 30-60-90 right triangle is a side of the equilateral triangle; therefore, each side length of the equilateral triangle is 2 k . The perimeter of a triangle is the sum of the lengths of each side. It's given that the perimeter of the equilateral triangle is 624 ; therefore, 2k+2k+2k=624, or 6k=624. Dividing both sides of this equation by 6 yields k = 104 .

Question 91 91 of 269 selected Area And Volume H

A right circular cone has a height of 22 centimeters (cm) and a base with a diameter of 6 cm. The volume of this cone is nπ cm3. What is the value of n ?

Show Answer Correct Answer: 66

The correct answer is 66 . It’s given that the right circular cone has a height of 22 centimeters (cm) and a base with a diameter of 6 cm. Since the diameter of the base of the cone is 6 cm, the radius of the base is 3 cm. The volume V , in cm3, of a right circular cone can be found using the formula V=13πr2h, where h is the height, in cm, and r is the radius, in cm, of the base of the cone. Substituting 22 for h and 3 for r in this formula yields V=13π(3)2(22), or V=66π. Therefore, the volume of the cone is 66π cm3. It’s given that the volume of the cone is nπ cm3. Therefore, the value of n is 66 .

Question 92 92 of 269 selected Right Triangles And Trigonometry H

In triangle A B C , angle B is a right angle. The length of side A B is 10 37 and the length of side B C is 24 37 . What is the length of side A C ?

  1. 14 37

  2. 26 37

  3. 34 37

  4. 34·37

Show Answer Correct Answer: B

Choice B is correct. The Pythagorean theorem states that for a right triangle, c2=a2+b2, where c represents the length of the hypotenuse and a and b represent the lengths of the legs. It’s given that in triangle ABC, angle B is a right angle. Therefore, triangle ABC is a right triangle, where the hypotenuse is side AC and the legs are sides AB and BC. It’s given that the lengths of sides AB and BC are 1037 and 2437, respectively. Substituting these values for a and b in the formula c2=a2+b2 yields c2=(1037)2+(2437)2, which is equivalent to c2=100(37)+576(37), or c2=676(37). Taking the square root of both sides of this equation yields c=±2637. Since c represents the length of the hypotenuse, side AC, c must be positive. Therefore, the length of side AC is 2637.

Choice A is incorrect. This is the result of solving the equation c=2437-1037, not c2=(1037)2+(2437)2.

Choice C is incorrect. This is the result of solving the equation c=1037+2437, not c2=(1037)2+(2437)2.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 93 93 of 269 selected Lines, Angles, And Triangles H
The figure presents right triangle M N P. Side M P is horizontal, with vertex M to the left of vertex P. Vertex N lies above side M P, and angle N is a right angle. Point Q lies on horizontal side M P directly below vertex N. Vertical line segment N Q is drawn, forming right triangles M N Q and Q N P. Side M N is labeled 3 and side N P is labeled 4.

In the figure above, what is the length of line segment NQ ?

  1. 2.2

  2. 2.3

  3. 2.4

  4. 2.5

Show Answer Correct Answer: C

Choice C is correct. First, line segment M P is the hypotenuse of right triangle M N P, whose legs have lengths 3 and 4. Therefore, open parenthesis, the length of line segment M P, close parenthesis, squared, equals, 3 squared plus 4 squared, so open parenthesis, the length of line segment M P, close parenthesis, squared, equals 25 and the length of line segment M P equals 5. Second, because angle M N P corresponds to angle N Q P and because angle M P N corresponds to angle N P Q, triangle M N P is similar to triangle N Q P. The ratio of corresponding sides of similar triangles is constant, so the length of line segment N Q over the length of line segment M N equals the length of line segment N P over the length of line segment M P. Since M P equals 5 and it’s given that the length of line segment M N equals 3 and the length of line segment N P equals 4, the length of line segment N Q, over 3, equals 4 over 5. Solving for NQ results in the length of line segment N Q equals 12 over 5, or 2.4.

Choices A, B, and D are incorrect and may result from setting up incorrect ratios.

 

Question 94 94 of 269 selected Area And Volume E

A rectangle has a length of 3 units and a width of 39 units. Which expression gives the area, in square units, of this rectangle?

  1. 2(3+39)

  2. 2(3·39)

  3. 3+39

  4. 3·39

Show Answer Correct Answer: D

Choice D is correct. The area of a rectangle is given by lw, where l is the length of the rectangle and w is the width of the rectangle. It's given that a rectangle has a length of 3 units and a width of 39 units. It follows that the area of the rectangle is 3·39 square units. Therefore, the expression that gives the area, in square units, of this rectangle, is 3·39.

Choice A is incorrect. This expression gives the perimeter, in units, of this rectangle.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 95 95 of 269 selected Lines, Angles, And Triangles H

Triangle X Y Z is similar to triangle R S T such that X , Y , and Z correspond to R , S , and T , respectively. The measure of Z is 20 ° and 2 X Y = R S . What is the measure of T?

  1. 2 °

  2. 10 °

  3. 20 °

  4. 40 °

Show Answer Correct Answer: C

Choice C is correct. It’s given that triangle X Y Z is similar to triangle R S T , such that X , Y , and Z correspond to R , S , and T , respectively. Since corresponding angles of similar triangles are congruent, it follows that the measure of Z is congruent to the measure of T. It’s given that the measure of Z is 20°. Therefore, the measure of T is 20°.

Choice A is incorrect and may result from a conceptual error.

Choice B is incorrect. This is half the measure of Z.

Choice D is incorrect. This is twice the measure of Z.

Question 96 96 of 269 selected Circles H

Point O is the center of a circle. The measure of arc R S on this circle is 100°. What is the measure, in degrees, of its associated angle ROS?

Show Answer Correct Answer: 100

The correct answer is 100 . It's given that point O is the center of a circle and the measure of arc RS on the circle is 100°. It follows that points R and S lie on the circle. Therefore, OR¯ and OS¯ are radii of the circle. A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle. Therefore, ROS is a central angle. Because the degree measure of an arc is equal to the measure of its associated central angle, it follows that the measure, in degrees, of ROS is 100 .

Question 97 97 of 269 selected Area And Volume E

The table gives the perimeters of similar triangles TUV and XYZ, where TU corresponds to XY. The length of TU is 6 .

  Perimeter
Triangle TUV 50
Triangle XYZ 150

What is the length of XY?

  1. 2

  2. 6

  3. 18

  4. 56

Show Answer Correct Answer: C

Choice C is correct. It’s given that triangle TUV is similar to triangle XYZ, and TU¯ corresponds to XY¯. If two triangles are similar, then the ratio of their perimeters is equal to the ratio of their corresponding sides. It’s given that the perimeter of triangle TUV is 50, the perimeter of triangle XYZ is 150, and the length of TU¯ is 6. Let n represent the length of XY¯. It follows that 50150=6n, or 13=6n. Multiplying each side of this equation by n yields n3=6. Multiplying each side of this equation by 3 yields n=18. Therefore, the length of XY¯ is 18.

Choice A is incorrect. This is the solution to 31=6n, not 13=6n.

Choice B is incorrect. This is the length of TU¯, not XY¯.

Choice D is incorrect. This is the sum of the length of TU¯ and the perimeter of triangle TUV, not the length of XY¯.

Question 98 98 of 269 selected Lines, Angles, And Triangles M

  • Clockwise from top left, the 3 lines are labeled m, r, and s.
  • Line m intersects both line r and line s.
  • At the intersection of line m and line r, 1 angle is labeled clockwise from top left as follows:
    • Top right: x°
  • At the intersection of line m and line s, 1 angle is labeled clockwise from top left as follows:
    • Bottom right: y°
  • A note indicates the figure is not drawn to scale.

In the figure shown, lines r and s are parallel, and line m intersects both lines. If y<65, which of the following must be true?

  1. x<115

  2. x>115

  3. x+y<180

  4. x+y>180

Show Answer Correct Answer: B

Choice B is correct. In the figure shown, the angle measuring y° is congruent to its vertical angle formed by lines s and m , so the measure of the vertical angle is also y°. The vertical angle forms a same-side interior angle pair with the angle measuring x°. It's given that lines r and s are parallel. Therefore, same-side interior angles in the figure are supplementary, which means the sum of the measure of the vertical angle and the measure of the angle measuring x° is 180°, or x + y = 180 . Subtracting x from both sides of this equation yields y=180-x. Substituting 180-x for y in the inequality y<65 yields 180-x<65. Adding x to both sides of this inequality yields 180<65+x. Subtracting 65 from both sides of this inequality yields 115<x, or x>115. Thus, if y<65, it must be true that x>115.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. x + y must be equal to, not less than, 180 .

Choice D is incorrect. x + y must be equal to, not greater than, 180 .

Question 99 99 of 269 selected Area And Volume M

A circle has a circumference of 31π centimeters. What is the diameter, in centimeters, of the circle?

Show Answer Correct Answer: 31

The correct answer is 31 . The circumference of a circle is equal to 2πr centimeters, where r represents the radius, in centimeters, of the circle, and the diameter of the circle is equal to 2 r centimeters. It's given that a circle has a circumference of 31π centimeters. Therefore, 31π=2πr. Dividing both sides of this equation by π yields 31 = 2 r . Since the diameter of the circle is equal to 2 r centimeters, it follows that the diameter, in centimeters, of the circle is 31 .

Question 100 100 of 269 selected Circles H

The equation open parenthesis, x plus 6, close parenthesis, squared, plus, open parenthesis, y plus 3, close parenthesis, squared, equals 121 defines a circle in the xy‑plane. What is the radius of the circle?

Show Answer

The correct answer is 11. A circle with equation open parenthesis, x minus a, close parenthesis, squared, plus, open parenthesis, y minus b, close parenthesis, squared, equals r squared, where a, b, and r are constants, has center with coordinates a, comma b and radius r. Therefore, the radius of the given circle is the square root of 121, or 11.

Question 101 101 of 269 selected Area And Volume M

Circle K has a radius of 4 millimeters (mm). Circle L has an area of 100π mm2. What is the total area, in mm2, of circles K and L ?

  1. 14π

  2. 28π

  3. 56π

  4. 116π

Show Answer Correct Answer: D

Choice D is correct. The area, A , of a circle is given by the formula A=πr2, where r represents the radius of the circle. It’s given that circle K has a radius of 4 millimeters (mm). Substituting 4 for r in the formula A=πr2 yields A=π(4)2, or A=16π. Therefore, the area of circle K is 16π mm2. It’s given that circle L has an area of 100π mm2. Therefore, the total area, in mm2, of circles K and L is 16π+100π, or 116π.

Choice A is incorrect. This is the sum of the radii, in mm, of circles K and L multiplied by π, not the total area, in mm2, of the circles.

Choice B is incorrect. This is the sum of the diameters, in mm, of circles K and L multiplied by π, not the total area, in mm2, of the circles.

Choice C is incorrect and may result from conceptual or calculation errors. 

Question 102 102 of 269 selected Circles M

A circle in the xy-plane has the equation (x-13)2+(y-k)2=64. Which of the following gives the center of the circle and its radius?

  1. The center is at (13,k) and the radius is 8 .

  2. The center is at (k,13) and the radius is 8 .

  3. The center is at (k,13) and the radius is 64 .

  4. The center is at (13,k) and the radius is 64 .

Show Answer Correct Answer: A

Choice A is correct. For a circle in the xy-plane that has the equation (x-h)2+(y-k)2=r2, where h , k , and r are constants, (h,k) is the center of the circle and the positive value of r is the radius of the circle. In the given equation, h = 13 and r 2 = 64 . Taking the square root of each side of r 2 = 64 yields r=±8. Therefore, the center of the circle is at (13,k) and the radius is 8 .

Choice B is incorrect. This gives the center and radius of a circle with equation (x-k)2+(y-13)2=64, not (x-13)2+(y-k)2=64.

Choice C is incorrect. This gives the center and radius of a circle with equation (x-k)2+(y-13)2=4,096 , not (x-13)2+(y-k)2=64.

Choice D is incorrect. This gives the center and radius of a circle with equation (x-13)2+(y-k)2=4,096 , not (x-13)2+(y-k)2=64.

Question 103 103 of 269 selected Lines, Angles, And Triangles M

Triangle A B C is similar to triangle X Y Z , such that A , B , and C correspond to X , Y , and Z respectively. The length of each side of triangle X Y Z is 2 times the length of its corresponding side in triangle A B C . The measure of side A B is 16. What is the measure of side X Y ?

  1. 14

  2. 16

  3. 18

  4. 32

Show Answer Correct Answer: D

Choice D is correct. It's given that triangle ABC is similar to triangle XYZ, such that A , B , and C correspond to X , Y , and Z , respectively. Therefore, side A B corresponds to side X Y . Since the length of each side of triangle XYZ is 2 times the length of its corresponding side in triangle ABC, it follows that the measure of side X Y is 2 times the measure of side A B . Thus, since the measure of side A B is 16 , then the measure of side X Y is 2(16), or 32 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the measure of side A B , not side X Y .

Choice C is incorrect and may result from conceptual or calculation errors.

Question 104 104 of 269 selected Lines, Angles, And Triangles E

In XYZ, the measure of X is 23 ° and the measure of Y is 66 °. What is the measure of Z?

  1. 43 °

  2. 89 °

  3. 91 °

  4. 179 °

Show Answer Correct Answer: C

Choice C is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is 180°. It's given that in XYZ, the measure of X is 23° and the measure of Y is 66°. It follows that the measure of Z is (180-23-66)°, or 91°.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the sum of the measures of X and Y, not the measure of Z.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 105 105 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the lines are labeled t, m, and n.
  • Line t intersects both line m and line n.
  • At the intersection of line t and line m, 1 angle is labeled clockwise from top left as follows:
    • Bottom left: x°
  • At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:
    • Bottom right: 33°
  • A note indicates the figure is not drawn to scale.

In the figure, line m is parallel to line n , and line t intersects both lines. What is the value of x

  1. 33

  2. 57

  3. 123

  4. 147

Show Answer Correct Answer: D

Choice D is correct. It’s given that line m is parallel to line n , and line t intersects both lines. It follows that line t is a transversal. When two lines are parallel and intersected by a transversal, exterior angles on the same side of the transversal are supplementary. Thus, x + 33 = 180 . Subtracting 33 from both sides of this equation yields x = 147 . Therefore, the value of x is 147 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 106 106 of 269 selected Lines, Angles, And Triangles M

In triangle JKL, the measures of K and L are each 48°. What is the measure of J, in degrees? (Disregard the degree symbol when entering your answer.)

Show Answer Correct Answer: 84

The correct answer is 84 . The sum of the measures of the interior angles of a triangle is 180°. It's given that in triangle JKL, the measures of Kand L are each 48°. Adding the measures, in degrees, of K and L gives 48+48, or 96 . Therefore, the measure of J, in degrees, is 180-96, or 84 .

Question 107 107 of 269 selected Lines, Angles, And Triangles E

In RST, the measure of R is 63°. Which of the following could be the measure, in degrees, of S?

  1. 116

  2. 118

  3. 126

  4. 180

Show Answer Correct Answer: A

Choice A is correct. The sum of the measures of the angles of a triangle is 180°. Therefore, the sum of the measures of R, S, and T is 180°. It's given that the measure of R is 63°. It follows that the sum of the measures of S and T is (180-63)°, or 117°. Therefore, the measure of S, in degrees, must be less than 117 . Of the given choices, only 116 is less than 117 . Thus, the measure, in degrees, of S could be 116 .

Choice B is incorrect. If the measure of S is 118°, then the sum of the measures of the angles of the triangle is greater than, not equal to, 180°.

Choice C is incorrect. If the measure of S is 126°, then the sum of the measures of the angles of the triangle is greater than, not equal to, 180°.

Choice D is incorrect. This is the sum of the measures of the angles of a triangle, in degrees.

Question 108 108 of 269 selected Circles H

A circle in the xy-plane has its center at (-4,-6). Line k is tangent to this circle at the point (-7,-7). What is the slope of line k ?

  1. -3

  2. - 1 3

  3. 1 3

  4. 3

Show Answer Correct Answer: A

Choice A is correct. A line that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It's given that the circle has its center at (-4,-6) and line k is tangent to the circle at the point (-7,-7). The slope of a radius defined by the points (q,r) and (s,t) can be calculated as t-rs-q. The points (-7,-7) and (-4,-6) define the radius of the circle at the point of tangency. Therefore, the slope of this radius can be calculated as (-6)-(-7)(-4)-(-7), or 13. If a line and a radius are perpendicular, the slope of the line must be the negative reciprocal of the slope of the radius. The negative reciprocal of 13 is -3. Thus, the slope of line k is -3.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the slope of the radius of the circle at the point of tangency, not the slope of line k .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 109 109 of 269 selected Area And Volume H

A manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the optimal number of containers to be displayed on a shelf. Which of the following expresses the volume, V, in cubic inches, of such containers, where r is the radius, in inches?

  1. V equals, 4 pi r cubed

  2. V equals, pi times, open parenthesis, 2 r, close parenthesis, cubed

  3. V equals, pi r squared, plus 4 pi r

  4. V equals, pi r cubed, plus 4 pi r squared

Show Answer Correct Answer: D

Choice D is correct. The volume, V, of a right cylinder is given by the formula V equals, pi r squared, times h, where r represents the radius of the base of the cylinder and h represents the height. Since the height is 4 inches longer than the radius, the expression r plus 4 represents the height of each cylindrical container. It follows that the volume of each container is represented by the equation V equals, pi r squared times, open parenthesis, r plus 4, close parenthesis. Distributing the expression pi r squared into each term in the parentheses yields V equals, pi r cubed, plus 4 pi r squared.

Choice A is incorrect and may result from representing the height as 4 r instead of r plus 4. Choice B is incorrect and may result from representing the height as 2 r instead of r plus 4. Choice C is incorrect and may result from representing the volume of a right cylinder as V equals pi r h instead of V equals, pi r squared, times h.

 

Question 110 110 of 269 selected Lines, Angles, And Triangles M

In triangle D E F , the measure of angle D is 47° and the measure of angle E is 97°. In triangle R S T , the measure of angle R is 47° and the measure of angle S is 97°. Which of the following additional pieces of information is needed to determine whether triangle D E F is similar to triangle R S T ?

  1. The measure of angle F

  2. The measure of angle T

  3. The measure of angle F and the measure of angle T

  4. No additional information is needed.

Show Answer Correct Answer: D

Choice D is correct. When two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. It's given that in triangle DEF, the measure of angle D is 47° and the measure of angle E is 97°. It's also given that in triangle RST, the measure of angle R is 47° and the measure of angle S is 97°. It follows that angle D is congruent to angle R and that angle E is congruent to angle S. Therefore, triangle DEF is similar to triangle RST and no additional information is needed.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Question 111 111 of 269 selected Area And Volume E

Rectangle P has an area of 72 square inches. If a rectangle with an area of 20 square inches is removed from rectangle P, what is the area, in square inches, of the resulting figure?

  1. 92

  2. 84

  3. 80

  4. 52

Show Answer Correct Answer: D

Choice D is correct. It's given that rectangle P has an area of 72 square inches. If a rectangle with an area of 20 square inches is removed from rectangle P, the area, in square inches, of the resulting figure is 72-20, or 52 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 112 112 of 269 selected Circles H

Circle A has equation ( x - 7 ) 2 + ( y + 3 ) 2 = 1 . In the x y -plane, circle B is obtained by translating circle A to the right 4 units. Which equation represents circle B?

  1. ( x - 7 ) 2 + ( y + 7 ) 2 = 1

  2. ( x - 3 ) 2 + ( y + 3 ) 2 = 1

  3. ( x - 11 ) 2 + ( y + 3 ) 2 = 1

  4. ( x - 7 ) 2 + ( y - 1 ) 2 = 1

Show Answer Correct Answer: C

Choice C is correct. The equation of a circle in the xy-plane can be written as (x-h)2+(y-k)2=r2, where the center of the circle is (h,k) and the radius of the circle is r units. It’s given that circle A has the equation (x-7)2+(y+3)2=1, which can be written as (x-7)2+(y-(-3))2=12. It follows that h=7, k=-3, and r=1. Therefore, the center of circle A is (7,-3) and its radius is 1 unit. If circle A is translated 4 units to the right, the x-coordinate of the center will increase by 4, while the y-coordinate and the radius of the circle will remain unchanged. Translating the center of circle A to the right 4 units yields (7+4,-3), or (11,-3). Therefore, the center of circle B is (11,-3). Substituting 11 for h, -3 for k, and 1 for r into the equation (x-h)2+(y-k)2=r2 yields (x-11)2+(y-(-3))2=1, or (x-11)2+(y+3)2=1. Therefore, the equation (x-11)2+(y+3)2=1 represents circle B.

Choice A is incorrect. This equation represents a circle obtained by shifting circle A down, rather than right, 4 units.

Choice B is incorrect. This equation represents a circle obtained by shifting circle A left, rather than right, 4 units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A up, rather than right, 4 units.

Question 113 113 of 269 selected Right Triangles And Trigonometry E

Triangle A B C is similar to triangle D E F , where angle A corresponds to angle D , and angles C and F are right angles. If cosB=122, what is the value of cosE?

  1. 1 22

  2. 1 23

  3. 21 22

  4. 22 23

Show Answer Correct Answer: A

Choice A is correct. The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to that angle to the length of the hypotenuse. It's given that angle C is a right angle in triangle ABC and that angle F is a right angle in triangle DEF. Therefore, cosB is equal to the ratio of the length of side BC to the length of side AB, and cosE is equal to the ratio of the length of side EF to the length of side DE. It’s also given that triangle ABC is similar to triangle DEF, where angle A corresponds to angle D. Since similar triangles have proportional side lengths, BCAB=EFDE. Therefore, the value of cosB is equal to the value of cosE. Since cosB=122, the value of cosE is 122.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 114 114 of 269 selected Right Triangles And Trigonometry H
The figure presents right triangle A, B C, with horizontal side A, C. Point B is directly above point A and angle A is a right angle. Point D lies on side A, B and point E lies on side B C. Horizontal line segment D E is drawn and angle D is a right angle.

In the figure above, tangent of B equals the fraction 3 over 4. If B C equals 15 and D A, equals 4, what is the length of line segment D E ?

Show Answer

The correct answer is 6. Since tangent of B equals three fourths, triangle A, B C and triangle D B E are both similar to 3-4-5 triangles. This means that they are both similar to the right triangle with sides of lengths 3, 4, and 5. Since the length of side B C equals 15, which is 3 times as long as the hypotenuse of the 3-4-5 triangle, the similarity ratio of triangle A, B C to the 3-4-5 triangle is 3:1. Therefore, the length of side A, C (the side opposite to angle B) is 3 times 3, equals 9, and the length of side A, B (the side adjacent to angle B) is 4 times 3, equals 12. It is also given that the length of side D A, equals 4. Since the length of side A, B equals, the length of side D A, plus the length of side D B and the length of side A, B equals 12, it follows that the length of side D B equals 8, which means that the similarity ratio of triangle D B E to the 3-4-5 triangle is 2:1 ( side D B is the side adjacent to angle B). Therefore, the length of side D E, which is the side opposite to angle B, is 3 times 2, equals 6.

Question 115 115 of 269 selected Lines, Angles, And Triangles M

Quadrilateral P'Q'R'S' is similar to quadrilateral P Q R S , where P , Q , R , and S correspond to P', Q', R', and S', respectively. The measure of angle P is 30°, the measure of angle Q is 50°, and the measure of angle R is 70°. The length of each side of P'Q'R'S' is 3 times the length of each corresponding side of P Q R S . What is the measure of angle P'?

  1. 10°

  2. 30°

  3. 40°

  4. 90°

Show Answer Correct Answer: B

Choice B is correct. It's given that quadrilateral P'Q'R'S' is similar to quadrilateral PQRS, where P , Q , R , and S correspond to P', Q', R', and S', respectively. Since corresponding angles of similar quadrilaterals are congruent, it follows that the measure of angle P is equal to the measure of angle P'. It's given that the measure of angle P is 30°. Therefore, the measure of angle P' is 30°.

Choice A is incorrect. This is 13 the measure of angle P'.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is 3 times the measure of angle P'.

Question 116 116 of 269 selected Lines, Angles, And Triangles H

A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of these eight angles is (7x-250)°. The sum of the measures of four of the eight angles is k°. Which of the following could NOT be equivalent to k , for all values of x ?

  1. -14x+1,540

  2. 14x-320

  3. -28x+1,720

  4. 360

Show Answer Correct Answer: A

Choice A is correct. It’s given that a line intersects two parallel lines, forming four acute angles and four obtuse angles. Since there are two parallel lines intersected by a transversal, all four acute angles have the same measure and all four obtuse angles have the same measure. Additionally, each acute angle is supplementary to each obtuse angle. It’s given that the measure of one of these eight angles is (7x-250)°. It follows that a supplementary angle has measure (180-(7x-250))°, or (-7x+430)°. It’s also given that the sum of the measures of four of the eight angles is k°. It follows that the possible values of k are 4(7x-250); (7x-250)+3(-7x+430); 2(7x-250)+2(-7x+430); 3(7x-250)+(-7x+430); and 4(-7x+430). These values are equivalent to 28x-1,000; -14x+1,040; 360; 14x-320; and -28x+1,720, respectively. It follows that of the given choices, only -14x+1,540 could NOT be equivalent to k, for all values of x.

Choice B is incorrect. This is the sum of three angles with measure (7x-250)° and one angle with measure (-7x+430)°.

Choice C is incorrect. This is the sum of four angles with measure (-7x+430)°.

Choice D is incorrect. This is the sum of two angles with measure (7x-250)° and two angles with measure (-7x+430)°.

Question 117 117 of 269 selected Circles M

The number of radians in a 720-degree angle can be written as a, times pi, where a is a constant. What is the value of a ?

Show Answer

The correct answer is 4. There are pi radians in a 180 degree angle. An angle measure of 720 degrees is 4 times greater than an angle measure of 180 degrees. Therefore, the number of radians in a 720 degree angle is 4 pi.

Question 118 118 of 269 selected Lines, Angles, And Triangles E

  • One angle is a right angle.
  • The measures of the other 2 angles are as follows:
    • 13°
  • A note indicates the figure is not drawn to scale.

In the right triangle shown, what is the value of a ?

  1. 13

  2. 77

  3. 90

  4. 103

Show Answer Correct Answer: B

Choice B is correct. The triangle shown is a right triangle, where the interior angle shown with a right angle symbol has a measure of 90°. It's shown that the other two interior angles measure 13° and a°. The sum of the measures of the interior angles of a triangle is 180°; therefore, 90+13+a=180. Combining like terms on the left-hand side of this equation yields 103+a=180. Subtracting 103 from both sides of this equation yields a = 77 .

Choice A is incorrect. This is the measure, in degrees, of the other acute interior angle of the right triangle, not the value of a .

Choice C is incorrect. This is the measure, in degrees, of the right angle of the right triangle, not the value of a .

Choice D is incorrect. This is the sum of the measures, in degrees, of the other two interior angles of the right triangle, not the value of a .

Question 119 119 of 269 selected Lines, Angles, And Triangles E

At a certain time and day, the Washington Monument in Washington, DC, casts a shadow that is 300 feet long. At the same time, a nearby cherry tree casts a shadow that is 16 feet long. Given that the Washington Monument is approximately 555 feet tall, which of the following is closest to the height, in feet, of the cherry tree?

  1. 10

  2. 20

  3. 30

  4. 35

Show Answer Correct Answer: C

Choice C is correct. There is a proportional relationship between the height of an object and the length of its shadow. Let c represent the height, in feet, of the cherry tree. The given relationship can be expressed by the proportion 555 over 300, equals, c over 16. Multiplying both sides of this equation by 16 yields c equals 29 point 6. This height is closest to the value given in choice C, 30.

Choices A, B, and D are incorrect and may result from calculation errors.

Question 120 120 of 269 selected Lines, Angles, And Triangles E
The figure presents line segments A, D and B E intersecting at point C. Line segments A, B and D E together with line segments A, D and B E form two triangles, A, B C and C D E.  In triangle A, B C, angle A, measures 20 degrees, angle B measures x degrees, and the measure of angle C is not given. In triangle C D E, angle D measures y degrees, angle E measures 40 degrees, and the measure of angle C is not given. Angle C of triangle A, B C appears to be congruent to angle C of triangle C D E. A note states that the figure is not drawn to scale.

In the figure above, line segment A, D intersects line segment B E at C. If x equals 100, what is the value of y ?

  1. 100

  2. 90

  3. 80

  4. 60

Show Answer Correct Answer: C

Choice C is correct. It’s given that x equals 100; therefore, substituting 100 for x in triangle ABC gives two known angle measures for this triangle. The sum of the measures of the interior angles of any triangle equals 180°. Subtracting the two known angle measures of triangle ABC from 180° gives the third angle measure: 180 degrees minus 100 degrees, minus 20 degrees, equals 60 degrees. This is the measure of angle BCA. Since vertical angles are congruent, the measure of angle DCE is also 60°. Subtracting the two known angle measures of triangle CDE from 180° gives the third angle measure: 180 degrees minus 60 degrees, minus 40 degrees, equals 80 degrees. Therefore, the value of y is 80.

Choice A is incorrect and may result from a calculation error. Choice B is incorrect and may result from classifying angle CDE as a right angle. Choice D is incorrect and may result from finding the measure of angle BCA or DCE instead of the measure of angle CDE.

Question 121 121 of 269 selected Right Triangles And Trigonometry M

  • One angle is a right angle.
  • The length of the side opposite the right angle is 21.
  • The length of one side adjacent to the right angle is 6.
  • The length of the other side adjacent to the right angle is a.
  • A note indicates the figure is not drawn to scale.

For the triangle shown, which expression represents the value of a ?

  1. 212-62

  2. 212-62

  3. 21-6

  4. 21-6

Show Answer Correct Answer: A

Choice A is correct. For the right triangle shown, the lengths of the legs are a units and 6 units, and the length of the hypotenuse is 21 units. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, a2+62=212. Subtracting 62 from both sides of this equation yields a2=212-62. Taking the square root of both sides of this equation yields a=±212-62. Since a is a length, a must be positive. Therefore, a=212-62. Thus, for the triangle shown, 212-62 represents the value of a .

Choice B is incorrect. For the triangle shown, this expression represents the value of a 2 , not a .

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 122 122 of 269 selected Circles E

What is the value of cos565π6?

  1. 12

  2. 1

  3. 32

  4. 3

Show Answer Correct Answer: C

Choice C is correct. The cosine of an angle is equal to the cosine of n(2π) radians more than the angle, where n is an integer constant. Since 565π6 is equivalent to 47(2π)+π6, cos(565π6) can be rewritten as cos(47(2π)+π6), which is equal to cos(π6). Therefore, the value of cos(565π6) is equal to the value of cos(π6), which is 32.

Alternate approach: A trigonometric ratio can be found using the unit circle, that is, a circle with radius 1 unit. The cosine of a number t is the x-coordinate of the point resulting from traveling a distance of t counterclockwise from the point (1,0) around a unit circle centered at the origin in the xy-plane. A unit circle has a circumference of 2π. It follows that since 565π6 is equal to 47(2π)+π6, traveling a distance of 565π6 counterclockwise around a unit circle means traveling around the circle completely 47 times and then another π6 beyond that. That is, traveling 565π6 results in the same point as traveling π6. Traveling π6 counterclockwise from the point (1,0) around a unit circle centered at the origin in the xy-plane results in the point (32,12). Thus, the value of cos565π6 is the x-coordinate of the point (32,12), which is 32.

Choice A is incorrect. This is the value of sin565π6, not cos565π6.

Choice B is incorrect. This is the value of the cosine of a multiple of 2π, not 565π6.

Choice D is incorrect. This is the value of 1tan565π6, not cos565π6.

Question 123 123 of 269 selected Circles M

  • Clockwise from top left, the following points are on the circle:
    • upper M
    • upper N
    • upper P

Points M, N, and P lie on the circle shown. On this circle, minor arc MN has a length of 39 centimeters and major arc MPN has a length of 195 centimeters. What is the circumference, in centimeters, of the circle shown?

  1. 39

  2. 156

  3. 195

  4. 234

Show Answer Correct Answer: D

Choice D is correct. Since the endpoints of minor arc MN and major arc MPN are the same, and the arcs together form a full circle, the sum of the lengths of these two arcs is equal to the circumference of the circle. It's given that the length of minor arc MN is 39 centimeters and the length of major arc MPN is 195 centimeters. Therefore, the circumference of the circle, in centimeters, is 39+195, or 234.

Choice A is incorrect. This is the length, in centimeters, of minor arc MN, not the circumference, in centimeters, of the circle.

Choice B is incorrect. This is the difference of the lengths of major arc MPN and minor arc MN, in centimeters.

Choice C is incorrect. This is the length, in centimeters, of major arc MPN, not the circumference, in centimeters, of the circle.

Question 124 124 of 269 selected Lines, Angles, And Triangles M

In triangle ABC, the measure of angle A is 50 degrees. If triangle ABC is isosceles, which of the following is NOT a possible measure of angle B ?

  1. 50 degrees

  2. 65 degrees

  3. 80 degrees

  4. 100 degrees

Show Answer Correct Answer: D

Choice D is correct. The sum of the three interior angles in a triangle is 180 degrees. It’s given that angle A measures 50 degrees. If angle B measured 100 degrees, the measure of angle C would be 180 degrees minus, open parenthesis, 50 degrees plus 100 degrees, close parenthesis, equals 30 degrees. Thus, the measures of the angles in the triangle would be 50 degrees, 100 degrees, and 30 degrees. However, an isosceles triangle has two angles of equal measure. Therefore, angle B can’t measure 100 degrees.

Choice A is incorrect. If angle B has measure 50 degrees, then angle C would measure 180 degrees minus, open parenthesis, 50 degrees plus 50 degrees, close parenthesis, equals 80 degrees, and 50 degrees, 50 degrees, and 80 degrees could be the angle measures of an isosceles triangle. Choice B is incorrect. If angle B has measure 65 degrees, then angle C would measure 180 degrees minus, open parenthesis, 65 degrees plus 50 degrees, close parenthesis, equals 65 degrees, and 50 degrees, 65 degrees, and 65 degrees could be the angle measures of an isosceles triangle. Choice C is incorrect. If angle B has measure 80 degrees, then angle C would measure 180 degrees minus, open parenthesis, 80 degrees plus 50 degrees, close parenthesis, equals 50 degrees, and 50 degrees, 80 degrees, and 50 degrees could be the angle measures of an isosceles triangle.

 

Question 125 125 of 269 selected Lines, Angles, And Triangles H
The figure presents triangle R T U, with R U horizontal, and U to the right of R. A point V is on R U, such that V is closer to R than it is to U. Vertex T is above R U. The angle at T is labeled 114 degrees. Side U T is extended to point S, above vertex R, and line segment S V is drawn. The angle to the right of S V and below S T is labeled 31 degrees. The angle to the left of S V and above R V is labeled x degrees.

In the figure above, R T equals, T U. What is the value of x ?

  1. 72

  2. 66

  3. 64

  4. 58

Show Answer Correct Answer: C

Choice C is correct. Since the length of side R T equals the length of side T U, it follows that triangle R T U is an isosceles triangle with base RU. Therefore, angle T R U and angle T U R are the base angles of an isosceles triangle and are congruent. Let the measures of both angle T R U and angle T U R be t degrees. According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is 180 degrees. Therefore, 114 degrees plus 2 t degrees, equals 180 degrees, so t equals 33.

Note that angle T U R is the same angle as angle S U V. Thus, the measure of angle S U V is 33 degrees. According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles. Therefore, x degrees is equal to the sum of the measures of angle V S U and angle S U V; that is, 31 degrees plus 33 degrees, equals 64 degrees. Thus, the value of x is 64.

Choice B is incorrect. This is the measure of angle S T R, but angle S T R is not congruent to angle S V R. Choices A and D are incorrect and may result from a calculation error.

 

Question 126 126 of 269 selected Lines, Angles, And Triangles M

  • Clockwise from top left, the 3 lines are labeled t, m, and n.
  • Line t intersects both line m and line n.
  • At the intersection of line t and line m, 2 angles are labeled clockwise from top left as follows:
    • Top right: x°
    • Bottom left: y°
  • At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:
    • Top left: z°
  • A note indicates the figure is not drawn to scale.

In the figure, lines m and n are parallel. If x=6k+13 and y=8k-29, what is the value of z ?

  1. 3

  2. 21

  3. 41

  4. 139

Show Answer Correct Answer: C

Choice C is correct. Vertical angles, which are angles that are opposite each other when two lines intersect, are congruent. The figure shows that lines t and m intersect. It follows that the angle with measure x° and the angle with measure y° are vertical angles, so x = y . It's given that x = 6 k + 13 and y = 8 k - 29 . Substituting 6 k + 13 for x and 8 k - 29 for y in the equation x = y yields 6 k + 13 = 8 k - 29 . Subtracting 6 k from both sides of this equation yields 13 = 2 k - 29 . Adding 29 to both sides of this equation yields 42 = 2 k , or 2 k = 42 . Dividing both sides of this equation by 2 yields k = 21 . It's given that lines m and n are parallel, and the figure shows that lines m and n are intersected by a transversal, line t . If two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary. It follows that the same-side interior angles with measures y° and z° are supplementary, so y + z = 180 . Substituting 8 k - 29 for y in this equation yields 8k-29+z=180. Substituting 21 for k in this equation yields 8(21)-29+z=180, or 139+z=180. Subtracting 139 from both sides of this equation yields z = 41 . Therefore, the value of z is 41 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the value of k , not z .

Choice D is incorrect. This is the value of x or y , not z .

Question 127 127 of 269 selected Area And Volume H

Parallelogram A B C D is similar to parallelogram P Q R S . The length of each side of parallelogram P Q R S is 2 times the length of its corresponding side of parallelogram A B C D . The area of parallelogram A B C D is 5 square centimeters. What is the area, in square centimeters, of parallelogram P Q R S ?

  1. 7

  2. 10

  3. 20

  4. 25

Show Answer Correct Answer: C

Choice C is correct. It’s given that parallelogram ABCD is similar to parallelogram PQRS. When two parallelograms are similar, if the scale factor between their corresponding side lengths is k, the scale factor between their areas is k2. It’s given that the length of each side of parallelogram PQRS is 2 times the length of its corresponding side of parallelogram ABCD, so the scale factor between their corresponding side lengths is 2. It follows that the scale factor between their areas is 22, or 4. It’s given that the area, in square centimeters, of parallelogram ABCD is 5. It follows that the area, in square centimeters, of parallelogram PQRS is 5(4), or 20.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 128 128 of 269 selected Right Triangles And Trigonometry H

An isosceles right triangle has a hypotenuse of length 58 inches. What is the perimeter, in inches, of this triangle?

  1. 29 2

  2. 58 2

  3. 58 + 58 2

  4. 58 + 116 2

Show Answer Correct Answer: C

Choice C is correct. Since the triangle is an isosceles right triangle, the two sides that form the right angle must be the same length. Let x be the length, in inches, of each of those sides. The Pythagorean theorem states that in a right triangle, a 2 + b 2 = c 2 , where c is the length of the hypotenuse and a and b are the lengths of the other two sides. Substituting x for a , x for b , and 58 for c in this equation yields x2+x2=582, or 2x2=582. Dividing each side of this equation by 2 yields x2=5822, or x2=2·5824. Taking the square root of each side of this equation yields two solutions: x=5822 and x=-5822. The value of x must be positive because it represents a side length. Therefore, x=5822, or x = 29 2 . The perimeter, in inches, of the triangle is 58+x+x, or 58+2x. Substituting 29 2 for x in this expression gives a perimeter, in inches, of 58+2(292), or 58 + 58 2 .

Choice A is incorrect. This is the length, in inches, of each of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.

Choice B is incorrect. This is the sum of the lengths, in inches, of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 129 129 of 269 selected Circles H

A circle has center O , and points R and S lie on the circle. In triangle O R S , the measure of ROS is 88°. What is the measure of RSO, in degrees? (Disregard the degree symbol when entering your answer.)

Show Answer Correct Answer: 46

The correct answer is 46 . It's given that O is the center of a circle and that points R and S lie on the circle. Therefore, OR¯ and OS¯ are radii of the circle. It follows that OR=OS. If two sides of a triangle are congruent, then the angles opposite them are congruent. It follows that the angles RSO and ORS, which are across from the sides of equal length, are congruent. Let x° represent the measure of RSO. It follows that the measure of ORS is also x°. It's given that the measure of ROS is 88°. Because the sum of the measures of the interior angles of a triangle is 180°, the equation x°+x°+88°=180°, or 2x+88=180, can be used to find the measure of RSO. Subtracting 88  from both sides of this equation yields 2x=92. Dividing both sides of this equation by 2 yields x=46. Therefore, the measure of RSO, in degrees, is 46 .

Question 130 130 of 269 selected Area And Volume H

Rectangles ABCD and EFGH are similar. The length of each side of EFGH is 6 times the length of the corresponding side of ABCD. The area of ABCD is 54 square units. What is the area, in square units, of EFGH?

  1. 9

  2. 36

  3. 324

  4. 1,944

Show Answer Correct Answer: D

Choice D is correct. The area of a rectangle is given by b h , where b is the length of the base of the rectangle and h is its height. Let x represent the length, in units, of the base of rectangle ABCD, and let y represent its height, in units. Substituting x for b and y for h in the formula b h yields x y . Therefore, the area, in square units, of ABCD can be represented by the expression x y . It’s given that the length of each side of EFGH is 6 times the length of the corresponding side of ABCD. Therefore, the length, in units, of the base of EFGH can be represented by the expression 6 x , and its height, in units, can be represented by the expression 6 y . Substituting 6 x for b and 6 y for h in the formula b h yields (6x)(6y), which is equivalent to 36 x y . Therefore, the area, in square units, of EFGH can be represented by the expression 36 x y . It’s given that the area of ABCD is 54 square units. Since x y represents the area, in square units, of ABCD, substituting 54 for x y in the expression 36 x y yields 36(54), or 1,944 . Therefore, the area, in square units, of EFGH is 1,944 .

Choice A is incorrect. This is the area of a rectangle where the length of each side of the rectangle is 16, not 6 , times the length of the corresponding side of ABCD.

Choice B is incorrect. This is the area of a rectangle where the length of each side of the rectangle is 23, not 6 , times the length of the corresponding side of ABCD.

Choice C is incorrect. This is the area of a rectangle where the length of each side of the rectangle is 6, not 6 , times the length of the corresponding side of ABCD.

Question 131 131 of 269 selected Right Triangles And Trigonometry H

A rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the circumference of the circle. The diagonal of the rectangle is twice the length of the shortest side of the rectangle. The area of the rectangle is 1,089 3 square units. What is the length, in units, of the diameter of the circle?

Show Answer Correct Answer: 66

The correct answer is 66 . It's given that each vertex of the rectangle lies on the circumference of the circle. Therefore, the length of the diameter of the circle is equal to the length of the diagonal of the rectangle. The diagonal of a rectangle forms a right triangle with the shortest and longest sides of the rectangle, where the shortest side and the longest side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. Let s represent the length, in units, of the shortest side of the rectangle. Since it's given that the diagonal is twice the length of the shortest side, 2 s represents the length, in units, of the diagonal of the rectangle. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b , then a2+b2=c2. Substituting s for a and 2 s for c in this equation yields s2+b2=(2s)2, or s2+b2=4s2. Subtracting s2 from both sides of this equation yields b2=3s2. Taking the positive square root of both sides of this equation yields b=s3. Therefore, the length, in units, of the rectangle’s longest side is s3. The area of a rectangle is the product of the length of the shortest side and the length of the longest side. The lengths, in units, of the shortest and longest sides of the rectangle are represented by s and s3, and it’s given that the area of the rectangle is 1,0893 square units. It follows that 1,0893=s(s3), or 1,0893=s23. Dividing both sides of this equation by 3 yields 1,089=s2. Taking the positive square root of both sides of this equation yields 33=s. Since the length, in units, of the diagonal is represented by 2 s , it follows that the length, in units, of the diagonal is 2(33), or 66 . Since the length of the diameter of the circle is equal to the length of the diagonal of the rectangle, the length, in units, of the diameter of the circle is 66 .

Question 132 132 of 269 selected Right Triangles And Trigonometry M

One leg of a right triangle has a length of 43.2 millimeters. The hypotenuse of the triangle has a length of 196.8 millimeters. What is the length of the other leg of the triangle, in millimeters?

  1. 43.2

  2. 120

  3. 192

  4. 201.5

Show Answer Correct Answer: C

Choice C is correct. The Pythagorean theorem states that for a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. It's given that one leg of a right triangle has a length of 43.2 millimeters. It's also given that the hypotenuse of the triangle has a length of 196.8 millimeters. Let b represent the length of the other leg of the triangle, in millimeters. Therefore, by the Pythagorean theorem, 43.22+b2=196.82, or 1,866.24+b2=38,730.24. Subtracting 1,866.24 from both sides of this equation yields b 2 = 36,864 . Taking the positive square root of both sides of this equation yields b = 192 . Therefore, the length of the other leg of the triangle, in millimeters, is 192 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 133 133 of 269 selected Area And Volume E

Triangle ABC and triangle DEF are similar triangles, where line segment A B and line segment D E are corresponding sides. If the length of line segment D E equals, 2 times the length of line segment A B and the perimeter of triangle ABC is 20, what is the perimeter of triangle DEF ?

  1. 10

  2. 40

  3. 80

  4. 120

Show Answer Correct Answer: B

Choice B is correct. Since triangles ABC and DEF are similar and the length of side D E equals, 2 times the length of side A B, the length of each side of triangle DEF is two times the length of its corresponding side in triangle ABC. Therefore, the perimeter of triangle DEF is two times the perimeter of triangle ABC. Since the perimeter of triangle ABC is 20, the perimeter of triangle DEF is 40.

Choice A is incorrect. This is half, not two times, the perimeter of triangle ABC. Choice C is incorrect. This is two times the perimeter of triangle DEF rather than two times the perimeter of triangle ABC. Choice D is incorrect. This is six times, not two times, the perimeter of triangle ABC.

Question 134 134 of 269 selected Circles H

In the xy-plane, the graph of 2 x squared, minus 6 x, plus 2 y squared, plus 2 y, equals 45 is a circle. What is the radius of the circle?

  1. 5

  2. 6.5

  3. square root of 40​​​​​​​

  4. square root of 50​​​​​​​

Show Answer Correct Answer: A

Choice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard form, open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared, where the ordered pair h comma k is the center of the circle and the radius of the circle is r. To do this, divide the original equation, 2 x squared, minus 6 x, plus 2 y squared, plus 2 y, equals 45, by 2 to make the leading coefficients of x squared and y squared each equal to 1: as follows: x squared, minus 3 x, plus y squared, plus y, equals 22 point 5. Then complete the square to put the equation in standard form. To do so, first rewrite x squared, minus 3 x, plus y squared, plus y, equals 22 point 5 as open parenthesis, x squared, minus 3 x, plus 2 point 2 5, close parenthesis, minus 2 point 2 5, plus, open parenthesis, y squared, plus y, plus 0 point 2 5, close parenthesis, minus 0 point 2 5, equals 22 point 5. Second, add 2.25 and 0.25 to both sides of the equation: open parenthesis, x squared, minus 3 x, plus 2 point 2 5, close parenthesis, plus, open parenthesis, y squared, plus y, plus 0 point 2 5, close parenthesis, equals 25. Since x squared, minus 3 x, plus 2 point 2 5, equals, open parenthesis, x minus 1 point 5, close parenthesis, squared, y squared, plus y, plus 0 point 2 5, equals, open parenthesis, y plus 0 point 5, close parenthesis, squared, and 25 equals 5 squared, it follows that open parenthesis, x minus 1 point 5, close parenthesis, squared, plus, open parenthesis, y plus 0 point 5, close parenthesis, squared, equals 5 squared. Therefore, the radius of the circle is 5.

Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the xy-plane.

 

Question 135 135 of 269 selected Right Triangles And Trigonometry H

The perimeter of an equilateral triangle is 852 centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is w3 centimeters. What is the value of w ?

Show Answer Correct Answer: 284/3, 94.66, 94.67

The correct answer is 284 3 .  Since the perimeter of a triangle is the sum of the lengths of its sides, and the given triangle is equilateral, the length of each side is 8523, or 284 , centimeters (cm). Right triangle A M O can be formed, where M is the midpoint of one of the triangle’s sides, A is one of this side’s endpoints, and O is the center of the circle. It follows that A M is 2842, or 142 , cm. Additionally, triangle A M O has angles measuring 30°60°, and 90°, where the measure of angle OMA is 90° and the measure of angle OAM is 30°. It follows that the length of side M O is half the length of hypotenuse A O , and the length of side A M is 3 times the length of side M O . It’s given that AO=w3 cm. Therefore, MO=w32 cm and AM=w332 cm, which is equivalent to AM=3w2 cm. Since A M = 142 cm, it follows that 3w2=142. Multiplying both sides of this equation by 2 yields 3 w = 284 . Dividing both sides of this equation by 3 yields w = 284 3 . Note that 284/3, 94.66, and 94.67 are examples of ways to enter a correct answer.

Question 136 136 of 269 selected Lines, Angles, And Triangles M

  • Line segment upper Q upper R is extended to point upper S.
  • A note indicates the figure is not drawn to scale.

 

In triangle P Q R QR is extended to point S . The measure of PQR is 132°, and the measure of PRS is 163°. What is the measure of QPR?

  1. 48°

  2. 31°

  3. 24°

  4. 17°

Show Answer Correct Answer: B

Choice B is correct. In the figure shown, since QS¯ is a line segment, the sum of the measures of PRS and PRQ is 180°. It's given that the measure of PRS is 163°. Thus, the measure of PRQ is (180-163)°, or 17°. The sum of the measures of the interior angles of a triangle is 180°. It's given that the measure of PQR is 132°. Therefore, the measure of QPR is (180-17-132)°, or 31°

Choice A is incorrect. This is the measure of the supplement of PQR, not the measure of QPR.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the measure of PRQ, not the measure of QPR.

Question 137 137 of 269 selected Right Triangles And Trigonometry H

An isosceles right triangle has a perimeter of 94 + 94 2 inches. What is the length, in inches, of one leg of this triangle?

  1. 47

  2. 47 2

  3. 94

  4. 94 2

Show Answer Correct Answer: B

Choice B is correct. It's given that the right triangle is isosceles. In an isosceles right triangle, the two legs have equal lengths, and the length of the hypotenuse is 2 times the length of one of the legs. Let l represent the length, in inches, of each leg of the isosceles right triangle. It follows that the length of the hypotenuse is l2 inches. The perimeter of a figure is the sum of the lengths of the sides of the figure. Therefore, the perimeter of the isosceles right triangle is l+l+l2 inches. It's given that the perimeter of the triangle is 94+942 inches. It follows that l+l+l2=94+942. Factoring the left-hand side of this equation yields (1+1+2)l=94+942, or (2+2)l=94+942. Dividing both sides of this equation by 2+2 yields l=94+9422+2. Rationalizing the denominator of the right-hand side of this equation by multiplying the right-hand side of the equation by 2-22-2 yields l=(94+942)(2-2)(2+2)(2-2). Applying the distributive property to the numerator and to the denominator of the right-hand side of this equation yields l=188-942+1882-9444-22+22-4. This is equivalent to l=9422, or l=472. Therefore, the length, in inches, of one leg of the isosceles right triangle is 47 2 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the length, in inches, of the hypotenuse.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 138 138 of 269 selected Lines, Angles, And Triangles H

Quadrilaterals PQRS and WXYZ are similar, where P , Q , and R correspond to W , X , and Y , respectively. The measure of S is 135 °, P S = 45 , and W Z = 9 . What is the measure of Z?

  1. 5 °

  2. 27 °

  3. 45 °

  4. 135 °

Show Answer Correct Answer: D

Choice D is correct. Corresponding angles in similar figures have equal measure. It's given that quadrilaterals PQRS and WXYZ are similar and that P, Q, and R correspond to W, X, and Y. It follows that S corresponds to Z. It's also given that the measure of S is 135°. Therefore, the measure of Z is 135°

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. This is the supplement of the measure of Z, not the measure of Z.

Question 139 139 of 269 selected Right Triangles And Trigonometry H

RS=440

ST=384

TR=584

The side lengths of right triangle R S T are given. Triangle R S T   is similar to triangle U V W , where S corresponds to V and T corresponds to W . What is the value of tanW?

  1. 48 73

  2. 55 73

  3. 48 55

  4. 55 48

Show Answer Correct Answer: D

Choice D is correct. The hypotenuse of triangle RST is the longest side and is across from the right angle. The longest side length given is 584 , which is the length of side TR. Therefore, the hypotenuse of triangle RST is side TR, so the right angle is angle S . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side, which is the side across from the angle, to the length of the adjacent side, which is the side closest to the angle that is not the hypotenuse. It follows that the opposite side of angle T is side RS and the adjacent side of angle T is side ST. Therefore, tanT=RSST. Substituting 440 for RS and 384 for ST in this equation yields tanT=440384. This is equivalent to tanT=5548. It’s given that triangle RST is similar to triangle UVW, where S corresponds to V and T corresponds to W . It follows that R corresponds to U . Therefore, the hypotenuse of triangle UVW is side WU, which means tanW=UVVW. Since the lengths of corresponding sides of similar triangles are proportional, RSST=UVVW. Therefore, tanW=UVVW is equivalent to tanW=RSST, or tanW=tanT. Thus, tanW=5548.

Choice A is incorrect. This is the value of cosW, not tanW.

Choice B is incorrect. This is the value of sinW, not tanW.

Choice C is incorrect. This is the value of 1tanW, not tanW.

Question 140 140 of 269 selected Right Triangles And Trigonometry E

  • One angle is a right angle.
  • The measure of a second angle is x°.
  • The length of the side opposite the right angle is 73.
  • The length of the side opposite the angle with measure x° is 63.
  • A note indicates the figure is not drawn to scale.

In the right triangle shown, what is the value of sinx°?

  1. 173

  2. 1073

  3. 6373

  4. 13673

Show Answer Correct Answer: C

Choice C is correct. The sine of an acute angle in a right triangle is the ratio of the length of the side opposite that angle to the length of the hypotenuse. In the right triangle shown, it's given that the length of the side opposite the angle with measure x° is 63 units and the length of the hypotenuse is 73 units. Therefore, the value of sinx° is 6373.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 141 141 of 269 selected Lines, Angles, And Triangles H
The figure presents right triangle A B C, where angle C is a right angle. Side A C is horizontal, such that vertex A is to the left of vertex C, and vertex B is directly above vertex C. A vertical line segment is drawn from a point on side A B to a point on side A C and divides side A C into two segments. The length of the segment between vertex A and the vertical line segment is labeled 5 and the length of the segment between the vertical line segment and vertex C is labeled 7. The vertical line segment is labeled x. Side B C is labeled y. A note states the figure is not drawn to scale.

The area of triangle ABC above is at least 48 but no more than 60. If y is an integer, what is one possible value of x ?

Show Answer

The correct answer is either the fraction 10 over 3, the fraction 15 over 4, or the fraction 25 over 6. The area of triangle ABC can be expressed as one half times, open parenthesis, 5 plus 7, close parenthesis, times y or 6 y. It’s given that the area of triangle ABC is at least 48 but no more than 60. It follows that 48 is less than or equal to 6 y, which is less than or equal to 60. Dividing by 6 to isolate y in this compound inequality yields 8 is less than or equal to y, which is less than or equal to 10. Since y is an integer, y equals 8, 9, or 10. In the given figure, the two right triangles shown are similar because they have two pairs of congruent angles: their respective right angles and angle A. Therefore, the following proportion is true: the fraction x over y equals the fraction 5 over 12. Substituting 8 for y in the proportion results in the fraction x over 8 equals the fraction 5 over 12. Cross multiplying and solving for x yields the fraction 10 over 3. Substituting 9 for y in the proportion results in the fraction x over 9 equals the fraction 5 over 12. Cross multiplying and solving for x yields the fraction 15 over 4. Substituting 10 for y in the proportion results in the fraction x over 10 equals the fraction 5 over 12. Cross multiplying and solving for x yields the fraction 25 over 6. Note that 10/3, 15/4, 25/6, 3.333, 3.75, 4.166, and 4.167 are examples of ways to enter a correct answer.

Question 142 142 of 269 selected Area And Volume H

A rectangular poster has an area of 360 square inches. A copy of the poster is made in which the length and width of the original poster are each increased by 20 %. What is the area of the copy, in square inches?

Show Answer Correct Answer: 2592/5, 518.4

The correct answer is 518.4 . It's given that the area of the original poster is 360 square inches. Let l represent the length, in inches, of the original poster, and let w represent the width, in inches, of the original poster. Since the area of a rectangle is equal to its length times its width, it follows that 360=lw. It's also given that a copy of the poster is made in which the length and width of the original poster are each increased by 20%. It follows that the length of the copy is the length of the original poster plus 20% of the length of the original poster, which is equivalent to l+20100l inches. This length can be rewritten as l+0.2l inches, or 1.2l inches. Similarly, the width of the copy is the width of the original poster plus 20% of the width of the original poster, which is equivalent to w+20100w inches. This width can be rewritten as w+0.2w inches, or 1.2w inches. Since the area of a rectangle is equal to its length times its width, it follows that the area, in square inches, of the copy is equal to (1.2l)(1.2w), which can be rewritten as (1.2)(1.2)(lw). Since 360=lw, the area, in square inches, of the copy can be found by substituting 360 for lw in the expression (1.2)(1.2)(lw), which yields (1.2)(1.2)(360), or 518.4 . Therefore, the area of the copy, in square inches, is 518.4 .

Question 143 143 of 269 selected Circles M

What is the center of the circle in the xy-plane defined by the equation ( x - 1 ) 2 + ( y + 7 ) 2 = 1 ?

  1. (-1,-7)

  2. (-1,7)

  3. (1,-7)

  4. (1,7)

Show Answer Correct Answer: C

Choice C is correct. The equation of a circle in the xy-plane can be written as (x-h)2+(y-k)2=r2, where the center of the circle is (h,k) and the radius of the circle is r. It's given that the circle in the xy-plane is defined by the equation (x-1)2+(y+7)2=1. This equation can be written as (x-1)2+(y-(-7))2=1. For this equation, it follows that h=1 and k=-7. Therefore, the center of the circle in the xy-plane defined by the given equation is (1,-7).

Choice A is incorrect. This is the center of the circle in the xy-plane that is defined by the equation (x+1)2+(y+7)2=1, not (x-1)2+(y+7)2=1.

Choice B is incorrect. This is the center of the circle in the xy-plane that is defined by the equation (x+1)2+(y-7)2=1, not (x-1)2+(y+7)2=1.

Choice D is incorrect. This is the center of the circle in the xy-plane that is defined by the equation (x-1)2+(y-7)2=1, not (x-1)2+(y+7)2=1.

Question 144 144 of 269 selected Area And Volume H

A right rectangular prism has a length of 28 centimeters (cm), a width of 15 cm, and a height of 16 cm. What is the surface area, in cm2, of the right rectangular prism?

Show Answer Correct Answer: 2216

The correct answer is 2,216 . The surface area of a prism is the sum of the areas of all its faces. A right rectangular prism consists of six rectangular faces, where opposite faces are congruent. It's given that this prism has a length of 28 cm, a width of 15 cm, and a height of 16 cm. Thus, for this prism, there are two faces with area (28)(15) cm2, two faces with area (28)(16) cm2, and two faces with area (15)(16) cm2. Therefore, the surface area, in cm2, of the right rectangular prism is 2(28)(15)+2(28)(16)+2(15)(16), or 2,216 .

Question 145 145 of 269 selected Circles H

A circle in the xy-plane has its center at (16,17) and has a radius of 7 k . Which equation represents this circle?

  1. (x-16)2+(y-17)2=49k

  2. (x-16)2+(y-17)2=49k2

  3. (x-16)2+(y-17)2=7k

  4. (x-16)2+(y-17)2=7k2

Show Answer Correct Answer: B

Choice B is correct. The equation of a circle in the xy-plane can be written as (x-h)2+(y-k)2=r2, where the center of the circle is (h,k) and the radius of the circle is r. It’s given that this circle has a center at (16,17) and a radius of 7k. Substituting 16 for h, 17 for k, and 7k for r in (x-h)2+(y-k)2=r2 yields (x-16)2+(y-17)2=(7k)2, or (x-16)2+(y-17)2=49k2. Therefore, the equation that represents this circle is (x-16)2+(y-17)2=49k2.

Choice A is incorrect. This equation represents a circle with radius 7k, not 7k.

Choice C is incorrect. This equation represents a circle with radius 7k, not 7k.

Choice D is incorrect. This equation represents a circle with radius 7k, not 7k.

Question 146 146 of 269 selected Area And Volume E

What is the area, in square centimeters, of a rectangle with a length of 34 centimeters (cm) and a width of 29 cm?

Show Answer Correct Answer: 986

The correct answer is 986 . The area, A , of a rectangle is given by A=lw, where l is the length of the rectangle and w is its width. It’s given that the length of the rectangle is 34 centimeters (cm) and the width is 29 cm. Substituting 34 for l and 29 for w in the equation A=lw yields A=(34)(29), or A=986. Therefore, the area, in square centimeters, of this rectangle is 986 .

Question 147 147 of 269 selected Circles M

An angle has a measure of 9π20 radians. What is the measure of the angle in degrees?

Show Answer Correct Answer: 81

The correct answer is 81 . The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by 180 degreesπ radians. Multiplying the given angle measure, 9π20 radians, by 180 degreesπ radians yields (9π20 radians)(180 degreesπ radians), which is equivalent to 81 degrees.

Question 148 148 of 269 selected Area And Volume E

The lengths of two sides of a triangle are 4 centimeters and 6 centimeters. If the perimeter of the triangle is 18 centimeters, what is the length, in centimeters, of the third side of this triangle?

  1. 2

  2. 8

  3. 10

  4. 24

Show Answer Correct Answer: B

Choice B is correct. The perimeter of a triangle is the sum of the lengths of all three of its sides. It's given that the lengths of two sides of a triangle are 4 centimeters and 6 centimeters. Let x represent the length, in centimeters, of the third side of this triangle. The sum of the lengths, in centimeters, of all three sides of the triangle can be represented by the expression 4+6+x. Since it’s given that the perimeter of the triangle is 18 centimeters, it follows that 4+6+x=18, or 10+x=18. Subtracting 10 from both sides of this equation yields x = 8 . Therefore, the length, in centimeters, of the third side of this triangle is 8 .

Choice A is incorrect. If the length of the third side of this triangle were 2 centimeters, the perimeter, in centimeters, of the triangle would be 4+6+2, or 12 , not 18 .

Choice C is incorrect. If the length of the third side of this triangle were 10 centimeters, the perimeter, in centimeters, of the triangle would be 4+6+10, or 20 , not 18 .

Choice D is incorrect. If the length of the third side of this triangle were 24 centimeters, the perimeter, in centimeters, of the triangle would be 4+6+24, or 34 , not 18 .

Question 149 149 of 269 selected Circles M

x 2 + 58 x + y 2 = 0

In the xy-plane, the graph of the given equation is a circle. What are the coordinates (x,y) of the center of the circle?  

  1. (0,29)

  2. (0,-29)

  3. (29,0)

  4. (-29,0)

Show Answer Correct Answer: D

Choice D is correct. It’s given that in the xy-plane, the graph of x2+58x+y2=0 is a circle. The equation of a circle in the xy-plane can be written as (x-h)2+(y-k)2=r2, where the coordinates of the center of the circle are (h,k) and the radius of the circle is r. By completing the square, the equation x2+58x+y2=0 can be rewritten as (x2+58x+(582)2)+y2=0+(582)2, or (x2+58x+841)+y2=841. This equation is equivalent to (x+29)2+y2=841, or (x-(-29))2+(y-0)2=841. Therefore, h is -29 and k is 0, and the coordinates (x,y) of the center of the circle are (-29,0).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 150 150 of 269 selected Area And Volume H

A right circular cone has a volume of one third, pi  cubic feet and a height of 9 feet. What is the radius, in feet, of the base of the cone?

  1. one third

  2. the fraction 1 over the square root of 3, end fraction

  3. the square root of 3

  4. 3

Show Answer Correct Answer: A

Choice A is correct. The equation for the volume of a right circular cone is V equals, one third pi r squared times h. It’s given that the volume of the right circular cone is one third pi cubic feet and the height is 9 feet. Substituting these values for V and h, respectively, gives one third pi equals, one third pi r squared, times 9. Dividing both sides of the equation by one third pi gives 1 equals, r squared times 9. Dividing both sides of the equation by 9 gives one ninth equals r squared. Taking the square root of both sides results in two possible values for the radius, the square root of one ninth or the negative of the square root of one ninth. Since the radius can’t have a negative value, that leaves the square root of one ninth as the only possibility. Applying the quotient property of square roots, the square root of the fraction a, over b, equals, the fraction the square root of a, over the square root of b, results in r equals, the fraction the square root of 1 over the square root of 9, or r equals one third.

Choices B and C are incorrect and may result from incorrectly evaluating the square root of one ninth. Choice D is incorrect and may result from solving r squared equals 9 instead of r squared equals one ninth.

 

Question 151 151 of 269 selected Lines, Angles, And Triangles E
The figure presents a triangle. The bottom side is horizontal, the left side slants upward and slightly to the left, and the right side slants downward and to the right. The bottom left angle of the triangle measures 2 b degrees. The top angle of the triangle measures 31 degrees. The bottom right angle of the triangle measures a, degrees.

In the triangle above, a, equals 45. What is the value of b ?

  1. 52

  2. 59

  3. 76

  4. 104

Show Answer Correct Answer: A

Choice A is correct. The sum of the measures of the three interior angles of a triangle is 180°. Therefore, 31 plus 2 b, plus a, equals 180. Since it’s given that a, equals 45, it follows that 31 plus 2 b, plus 45, equals 180, or 2 b equals 104. Dividing both sides of this equation by 2 yields b equals 52.

Choice B is incorrect and may result from a calculation error. Choice C is incorrect. This is the value of a, plus 31 . Choice D is incorrect. This is the value of 2 b.

Question 152 152 of 269 selected Circles M

An angle has a measure of 16 π 15 radians. What is the measure of the angle, in degrees?

 

Show Answer Correct Answer: 192

The correct answer is 192 . The measure of an angle, in degrees, can be found by multiplying its measure, in radians, by 180 degreesπ radians. Multiplying the given angle measure, 16π15radians, by 180 degreesπ radians yields (16π15radians)(180 degreesπ radians), which simplifies to 192 degrees.

Question 153 153 of 269 selected Circles H

  • The circle passes through the following approximate points:
    • (0 comma 8.6)
    • (2.6 comma 6)
    • (0 comma 3.4)
    • (negative 2.6 comma 6)

Circle A shown is defined by the equation x2+(y-6)2=7. Circle B (not shown) has the same radius but is translated 96 units to the right. If the equation of circle B is (x-h)2+(y-k)2=a, where h , k , and a are constants, what is the value of 4 a ?

Show Answer Correct Answer: 28

The correct answer is 28. The equation of a circle in the xy-plane can be written as (x-t)2+(y-s)2=r2, where the center of the circle is (t,s) and the radius of the circle is r. It’s given that circle A is defined by the equation x2+(y-6)2=7, which can be written as (x-0)2+(y-6)2=(7)2. It follows that r=7 and the radius of circle A is 7. It’s also given that circle B has the same radius as circle A. If the equation of circle B is (x-h)2+(y-k)2=a, then a=r2. Substituting 7 for r in this equation yields a=(7)2, or a=7. It follows that the value of 4a is 4(7), or 28.

Question 154 154 of 269 selected Right Triangles And Trigonometry H

A right triangle has legs with lengths of 24 centimeters and 21 centimeters. If the length of this triangle's hypotenuse, in centimeters, can be written in the form 3d, where d is an integer, what is the value of d ?

Show Answer Correct Answer: 113

The correct answer is 113 . It's given that the legs of a right triangle have lengths 24 centimeters and 21 centimeters. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. It follows that if h represents the length, in centimeters, of the hypotenuse of the right triangle, h2=242+212. This equation is equivalent to h 2 = 1,017 . Taking the square root of each side of this equation yields h=1,017. This equation can be rewritten as h=9·113, or h=9·113. This equation is equivalent to h = 3 113 . It's given that the length of the triangle's hypotenuse, in centimeters, can be written in the form 3 d . It follows that the value of d is 113 .

Question 155 155 of 269 selected Right Triangles And Trigonometry H

RS=20

ST=48

TR=52

The side lengths of right triangle RST are given. Triangle RST is similar to triangle UVW, where S corresponds to V and T corresponds to W . What is the value of tanW?

  1. 5 13

  2. 5 12

  3. 12 13

  4. 12 5

Show Answer Correct Answer: B

Choice B is correct. It's given that right triangle R S T is similar to triangle U V W , where S corresponds to V and T corresponds to W . It's given that the side lengths of the right triangle R S T are R S = 20 , S T = 48 , and TR=52. Corresponding angles in similar triangles are equal. It follows that the measure of angle T is equal to the measure of angle W . The hypotenuse of a right triangle is the longest side. It follows that the hypotenuse of triangle RST is side TR. The hypotenuse of a right triangle is the side opposite the right angle. Therefore, angle S is a right angle. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle T is side S T . The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle T is side R S . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, tan T=RSST. Substituting 20 for R S and 48 for S T in this equation yields tan T=2048, or tan T=512. The tangents of two acute angles with equal measures are equal. Since the measure of angle T is equal to the measure of angle W , it follows that tan T=tan W. Substituting 512 for tan T in this equation yields 512=tan W. Therefore, the value of tan W is 512.

Choice A is incorrect. This is the value of sin W.

Choice C is incorrect. This is the value of cos W.

Choice D is incorrect. This is the value of 1tan W.

Question 156 156 of 269 selected Lines, Angles, And Triangles E

  • At the intersection of the 2 lines, the angles are labeled clockwise from left as follows:
    • Left: 1
    • Top: unlabeled
    • Right: 2
    • Bottom: unlabeled
  • A note indicates the figure is not drawn to scale.

In the figure, two lines intersect at a point. Angle 1 and angle 2 are vertical angles. The measure of angle 1 is 72 °. What is the measure of angle 2 ?

  1. 72 °

  2. 108 °

  3. 144 °

  4. 288 °

Show Answer Correct Answer: A

Choice A is correct. It’s given that angle 1 and angle 2 are vertical angles, and the measure of angle 1 is 72°. Vertical angles have equal measures. Therefore, the measure of angle 2 is 72°.

Choice B is incorrect. This is the measure of an angle that is supplementary, not congruent, to angle 1 .

Choice C is incorrect. This is the sum of the measures of angle 1 and angle 2 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 157 157 of 269 selected Lines, Angles, And Triangles E
The figure presents 3 lines intersecting at point P. Six angles are formed at the point of intersection. Of the six angles, one angle is labeled x degrees, and counterclockwise angles are labeled as follows: unlabeled, y degrees, unlabeled, z degrees, and unlabeled. A note states that the figure is not drawn to scale.

In the figure, three lines intersect at point P. If x equals 65 and y equals 75, what is the value of z ?

  1. 140

  2. 80

  3. 40

  4. 20

Show Answer Correct Answer: C

Choice C is correct. The angle that is shown as lying between the y° angle and the z° angle is a vertical angle with the x° angle. Since vertical angles are congruent and x equals 65, the angle between the y° angle and the z° angle measures 65°. Since the 65° angle, the y° angle, and the z° angle are adjacent and form a straight angle, it follows that the sum of the measures of these three angles is 180°, which is represented by the equation 65 degrees, plus y degrees, plus z degrees, equals 180 degrees. It’s given that y = 75. Substituting 75 for y yields 65 degrees, plus 75 degrees, plus z degrees, equals 180 degrees, which can be rewritten as 140 degrees, plus z degrees, equals 180 degrees. Subtracting 140° from both sides of this equation yields z degrees equals 40 degrees. Therefore, z equals 40.

Choice A is incorrect and may result from finding the value of x plus y rather than z. Choices B and D are incorrect and may result from conceptual or computational errors.

Question 158 158 of 269 selected Area And Volume M

A circle has a radius of 2.1 inches. The area of the circle is bπ square inches, where b is a constant. What is the value of b ?

Show Answer Correct Answer: 4.41, 441/100

The correct answer is 4.41 . The area, A , of a circle is given by the formula A=πr2, where r is the radius of the circle. It's given that the area of the circle is bπ square inches, where b is a constant, and the radius of the circle is 2.1 inches. Substituting bπ for A and 2.1 for r in the formula A=πr2 yields bπ=π(2.12). Dividing both sides of this equation by π yields b = 4.41 . Therefore, the value of b is 4.41 .

Question 159 159 of 269 selected Area And Volume E

A rectangle has a length of 13 and a width of 6 . What is the perimeter of the rectangle?

  1. 12

  2. 26

  3. 38

  4. 52

Show Answer Correct Answer: C

Choice C is correct. The perimeter of a quadrilateral is the sum of the lengths of its four sides. It's given that the rectangle has a length of 13 and a width of 6 . It follows that the rectangle has two sides with length 13 and two sides with length 6 . Therefore, the perimeter of the rectangle is 13+13+6+6, or 38 .

Choice A is incorrect. This is the sum of the lengths of the two sides with length 6 , not the sum of the lengths of all four sides of the rectangle.

Choice B is incorrect. This is the sum of the lengths of the two sides with length 13 , not the sum of the lengths of all four sides of the rectangle.

Choice D is incorrect. This is the perimeter of a rectangle that has four sides with length 13 , not two sides with length 13 and two sides with length 6 .

Question 160 160 of 269 selected Right Triangles And Trigonometry H

The figure presents right triangle R S T such that side R T is horizontal, vertex T is to the right of vertex R, and vertex S is above R T. Side R S is labeled 12. Side S T is labeled 5. Angle S is a right angle

In triangle RST above, point W (not shown) lies on line segment R T. What is the value of cosine of angle R S W, minus sine of angle W S T ?

Show Answer

The correct answer is 0. Note that no matter where point W is on side R T, the sum of the measures of angle R S W and angle W S T is equal to the measure of angle R S T, which is 90 degrees. Thus, angle R S W and angle W S T are complementary angles. Since the cosine of an angle is equal to the sine of its complementary angle, the cosine of angle R S W, equals, the sine of angle W S T. Therefore, the cosine of angle R S W, minus, the sine of angle W S T, equals 0.

Question 161 161 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled l, m, and n.
  • Line l intersects both line m and line n.
  • At the intersection of line l and line m, 1 angle is labeled clockwise from top left as follows:
    • Top right: x°
  • At the intersection of line l and line n, 1 angle is labeled clockwise from top left as follows:
    • Top left: 26°
  • A note indicates the figure is not drawn to scale.

In the figure shown, line m is parallel to line n . What is the value of x ?

  1. 13

  2. 26

  3. 52

  4. 154

Show Answer Correct Answer: D

Choice D is correct. The sum of consecutive interior angles between two parallel lines and on the same side of the transversal is 180 degrees. Since it's given that line m is parallel to line n , it follows that x + 26 = 180 . Subtracting 26 from both sides of this equation yields 154 . Therefore, the value of x is 154 .

Choice A is incorrect. This is half of the given angle measure.

Choice B is incorrect. This is the value of the given angle measure.

Choice C is incorrect. This is twice the value of the given angle measure.

Question 162 162 of 269 selected Right Triangles And Trigonometry M

A right triangle has legs with lengths of 28 centimeters and 20 centimeters. What is the length of this triangle's hypotenuse, in centimeters?

  1. 8 6

  2. 4 74

  3. 48

  4. 1,184

Show Answer Correct Answer: B

Choice B is correct. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. It's given that the right triangle has legs with lengths of 28 centimeters and 20 centimeters. Let c represent the length of this triangle's hypotenuse, in centimeters. Therefore, by the Pythagorean theorem, 282+202=c2, or 1,184=c2. Taking the positive square root of both sides of this equation yields 1,184=c, or 474=c. Therefore, the length of this triangle's hypotenuse, in centimeters, is 474.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the square of the length of the triangle’s hypotenuse.

Question 163 163 of 269 selected Lines, Angles, And Triangles E
The figure presents 2 horizontal lines l and k, with line l above line k. A third line, labeled j, runs from the bottom left and slants to the top right, intersecting both lines l and k. The angle above line l and to the left of line j is labeled a degrees. The angle above line k and to the right of line j is labeled 64 degrees. Note that the figure is not drawn to scale.

In the figure above, lines l and k are parallel. What is the value of a ?

  1. 26

  2. 64

  3. 116

  4. 154

Show Answer Correct Answer: C

Choice C is correct. Since lines l and k are parallel, corresponding angles formed by the intersection of line j with lines l and k are congruent. Therefore, the angle with measure a° must be the supplement of the angle with measure 64°. The sum of two supplementary angles is 180°, so a = 180 – 64 = 116.

Choice A is incorrect and likely results from thinking the angle with measure is the complement of the angle with measure 64°. Choice B is incorrect and likely results from thinking the angle with measure a° is congruent to the angle with measure 64°. Choice D is incorrect and likely results from a conceptual or computational error.

Question 164 164 of 269 selected Lines, Angles, And Triangles E

In triangle A B C , the measure of angle B is 52° and the measure of angle C is 17°. What is the measure of angle A ?

  1. 21 °

  2. 35 °

  3. 69 °

  4. 111 °

Show Answer Correct Answer: D

Choice D is correct. The sum of the angle measures of a triangle is 180°. Adding the measures of angles B and C gives 52+17=69°. Therefore, the measure of angle A is 180-69=111°.

Choice A is incorrect and may result from subtracting the sum of the measures of angles B and C from 90°, instead of from 180°.

Choice B is incorrect and may result from subtracting the measure of angle C from the measure of angle B .

Choice C is incorrect and may result from adding the measures of angles B and C but not subtracting the result from 180°.

Question 165 165 of 269 selected Circles E
The figure presents a circle with center O. Points A, and C are indicated on the circle, creating minor arc A, C. A diameter is drawn from point A to a point on the other side of the circle. Similarly, a diameter is drawn from point C to a point on the other side of the circle. The two lines intersect at the origin, forming a right angle at angle A, O C

The circle above with center O has a circumference of 36. What is the length of minor arc A, C?

  1. 9

  2. 12

  3. 18

  4. 36

Show Answer Correct Answer: A

Choice A is correct. A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and angle A, O C is a central angle of the circle. From the figure, the two diameters that meet to form angle A, O C are perpendicular, so the measure of angle A, O C is 90 degrees. Therefore, the length of minor arc A, C is the fraction 90 over 360 of the circumference of the circle. Since the circumference of the circle is 36, the length of minor arc A, C is the fraction 90 over 360, end fraction, times 36, equals 9.

Choices B, C, and D are incorrect. The perpendicular diameters divide the circumference of the circle into four equal arcs; therefore, minor arc A, C is one fourth of the circumference. However, the lengths in choices B and C are, respectively, one third and one half the circumference of the circle, and the length in choice D is the length of the entire circumference. None of these lengths is one fourth the circumference.

 

Question 166 166 of 269 selected Circles M

The measure of angle R is 2 π 3 radians. The measure of angle T is 5 π 12 radians greater than the measure of angle R . What is the measure of angle T , in degrees?

  1. 75

  2. 120

  3. 195

  4. 390

Show Answer Correct Answer: C

Choice C is correct. It’s given that the measure of angle R is 2π3 radians, and the measure of angle T is 5π12 radians greater than the measure of angle R . Therefore, the measure of angle T is equal to 2π3+5π12 radians. Multiplying 2π3 by 44 to get a common denominator with 5π12 yields 8π12. Therefore, 2π3+5π12 is equivalent to 8π12+5π12, or 13π12. Therefore, the measure of angle T is 13π12 radians. The measure of angle T , in degrees, can be found by multiplying its measure, in radians, by 180π. This yields 13π12×180π, which is equivalent to 195 degrees. Therefore, the measure of angle T is 195 degrees.

Choice A is incorrect. This is the number of degrees that the measure of angle T is greater than the measure of angle R .

Choice B is incorrect. This is the measure of angle R , in degrees.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 167 167 of 269 selected Right Triangles And Trigonometry H

Triangle A B C is similar to triangle D E F , where A corresponds to D and C corresponds to F . Angles C and F are right angles. If tan(A)=3 and DF=125, what is the length of DE¯?

  1. 12533

  2. 12532

  3. 1253

  4. 250

Show Answer Correct Answer: D

Choice D is correct. Corresponding angles in similar triangles have equal measures. It's given that triangle ABC is similar to triangle DEF, where A corresponds to D , so the measure of angle A is equal to the measure of angle D . Therefore, if tan(A)=3, then tan(D)=3. It's given that angles C and F are right angles, so triangles ABC and DEF are right triangles. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle D is side D F . The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle D is side E F . The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, tan(D)=EFDF. If DF=125, the length of side E F can be found by substituting 3 for tan(D) and 125 for D F in the equation tan(D)=EFDF, which yields 3=EF125. Multiplying both sides of this equation by 125 yields 1253=EF. Since the length of side E F is 3 times the length of side D F , it follows that triangle DEF is a special right triangle with angle measures 30°, 60°, and 90°. Therefore, the length of the hypotenuse, DE¯, is 2 times the length of side D F , or DE=2(DF). Substituting 125 for D F in this equation yields DE=2(125), or DE=250. Thus, if tan(A)=3 and DF=125, the length of DE¯ is 250 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the length of EF¯, not DE¯.

Question 168 168 of 269 selected Lines, Angles, And Triangles M
The figure presents two triangles, triangle A, B C and triangle D E F. The two triangles are placed side by side, and triangle A, B C is larger than triangle D E F. In triangle A, B C, side A, B slants upward and to the right, side B C slants downward and to the right, and side A, C slants slightly upward and to the right. In triangle D E F, side D E slants upward and to the right, side E F slants downward and to the right, and side D F slants slightly upward and to the right. A note indicates that the figures are not drawn to scale.

Triangle ABC and triangle DEF are shown. The relationship between the side lengths of the two triangles is such that the length of side A, B, over, the length of side D E, equals, the length of side B C, over, the length of side E F, which equals, the length of side A, C, over, the length of side D F, which equals 3. If the measure of angle BAC is 20°, what is the measure, in degrees, of angle EDF ? (Disregard the degree symbol when gridding your answer.)

Show Answer

The correct answer is 20. By the equality given, the three pairs of corresponding sides of the two triangles are in the same proportion. By the side-side-side (SSS) similarity theorem, triangle ABC is similar to triangle DEF. In similar triangles, the measures of corresponding angles are congruent. Since angle BAC corresponds to angle EDF, these two angles are congruent and their measures are equal. It’s given that the measure of angle BAC is 20°, so the measure of angle EDF is also 20°.

Question 169 169 of 269 selected Circles M

The measure of angle Z is 60 °. What is the measure, in radians, of angle Z ?

  1. 1 6 π

  2. 1 3 π

  3. 2 3 π

  4. 1 π

Show Answer Correct Answer: B

Choice B is correct. The measure of an angle, in radians, can be found by multiplying its measure, in degrees, by π180. It's given that the measure of angle Z is 60°. It follows that the measure, in radians, of angle Z is 60(π180), or 13π.

Choice A is incorrect. This is the measure, in radians, of an angle whose measure is 30°, not 60°.

Choice C is incorrect. This is the measure, in radians, of an angle whose measure is 120°, not 60°.

Choice D is incorrect. This is the measure, in radians, of an angle whose measure is 180°, not 60°.

Question 170 170 of 269 selected Circles H

The equation x 2 + ( y - 1 ) 2 = 49 represents circle A. Circle B is obtained by shifting circle A down 2 units in the xy-plane. Which of the following equations represents circle B?

  1. ( x - 2 ) 2 + ( y - 1 ) 2 = 49

  2. x 2 + ( y - 3 ) 2 = 49

  3. ( x + 2 ) 2 + ( y - 1 ) 2 = 49

  4. x 2 + ( y + 1 ) 2 = 49

Show Answer Correct Answer: D

Choice D is correct. The graph in the xy-plane of an equation of the form (x-h)2+(y-k)2=r2 is a circle with center (h,k) and a radius of length r . It's given that circle A is represented by x2+(y-1)2=49, which can be rewritten as x2+(y-1)2=72. Therefore, circle A has center (0,1) and a radius of length 7 . Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point (x,y) on circle A, the point (x,y-2) lies on circle B. Moreover, if (h,k) is the center of circle A, then (h,k-2) is the center of circle B. Therefore, circle B has a radius of 7 and the center of circle B is (0,1-2), or (0,-1). Thus, circle B can be represented by the equation x2+(y+1)2=72, or x2+(y+1)2=49.

Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right 2 units.

Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up 2 units.

Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left 2 units.

Question 171 171 of 269 selected Area And Volume M

  • Rectangle upper A upper B upper C upper D is labeled as follows:
    • The length of side upper A upper B is 12 inches.
  • Rectangle upper Q upper R upper S upper T is labeled as follows:
    • The length of side upper Q upper R is 5 inches.
    • The length of side upper Q upper T is 10 inches.
  • A note indicates the figure is not drawn to scale.

 

Rectangles A B C D and Q R S T shown are similar, where A , B , C , and D correspond to Q , R , S , and T , respectively. What is the length, in inches (in), of AD¯?

  1. 60

  2. 24

  3. 17

  4. 10

Show Answer Correct Answer: B

Choice B is correct. It’s given that rectangles ABCD and QRST are similar, where A, B, C, and D correspond to Q, R, S, and T, respectively. It follows that AB¯ corresponds to QR¯ and AD¯ corresponds to QT¯. If two rectangles are similar, then the lengths of their corresponding sides are proportional. It’s given in the figure that the length of AB¯ is 12 inches, the length of QR¯ is 5 inches, and the length of QT¯ is 10 inches. If x is the length, in inches, of AD¯, then 125 is equivalent to x10. Therefore, the value of x can be found using the equation 125=x10. Multiplying each side of this equation by 10 yields 1205=x, or 24=x. Therefore, the length, in inches, of AD¯ is 24.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the length, in inches, of QT¯, not AD¯.

Question 172 172 of 269 selected Lines, Angles, And Triangles H

In triangle X Y Z , angle Y is a right angle, point P lies on XZ¯, and point Q lies on YZ¯ such that PQ¯ is parallel to XY¯. If the measure of angle XZY is 63°, what is the measure, in degrees, of angle XPQ?

Show Answer Correct Answer: 153

The correct answer is 153 . Since it's given that PQ¯ is parallel to XY¯ and angle Y is a right angle, angle ZQP is also a right angle. Angle ZPQ is complementary to angle XZY, which means its measure, in degrees, is 90-63, or 27 . Since angle XPQ is supplementary to angle ZPQ, its measure, in degrees, is 180-27, or 153 .

Question 173 173 of 269 selected Circles H

What is the value of tan92π3?

  1. - 3

  2. - 3 3

  3. 3 3

  4. 3

Show Answer Correct Answer: A

Choice A is correct. A trigonometric ratio can be found using the unit circle, that is, a circle with radius 1 unit. If a central angle of a unit circle in the xy-plane centered at the origin has its starting side on the positive x-axis and its terminal side intersects the circle at a point (x,y), then the value of the tangent of the central angle is equal to the y-coordinate divided by the x-coordinate. There are 2 π radians in a circle. Dividing 92 π 3 by 2 π yields 926, which is equivalent to 15+23. It follows that on the unit circle centered at the origin in the xy-plane, the angle 92 π 3 is the result of 15 revolutions from its starting side on the positive x-axis followed by a rotation through 2 π 3 radians. Therefore, the angles 92 π 3 and 2 π 3 are coterminal angles and tan(92π3) is equal to tan(2π3). Since 2 π 3 is greater than π 2 and less than π , it follows that the terminal side of the angle is in quadrant II and forms an angle of π 3 , or 60°, with the negative x-axis. Therefore, the terminal side of the angle intersects the unit circle at the point (-12,32). It follows that the value of tan(2π3) is 32-12, which is equivalent to - 3 . Therefore, the value of tan(92π3) is - 3 .

Choice B is incorrect. This is the value of 1tan(92π3), not tan(92π3).

Choice C is incorrect. This is the value of 1tan(π3), not tan(92π3).

Choice D is incorrect. This is the value of tan(π3), not tan(92π3).

Question 174 174 of 269 selected Lines, Angles, And Triangles H

Triangles P Q R and L M N are graphed in the xy-plane. Triangle P Q R has vertices P , Q , and R at (4,5)(4,7), and (6,5), respectively. Triangle L M N has vertices L , M , and N at (4,5)(4,7+k), and (6+k,5), respectively, where k is a positive constant. If the measure of Q is t°, what is the measure of N?

  1. (90-(t-k))°

  2. (90-(t+k))°

  3. (90-t)°

  4. (90+k)°

Show Answer Correct Answer: C

Choice C is correct. Since P=(4,5) and Q=(4,7), side P Q is parallel to the y-axis and has a length of 2 . Since P=(4,5) and R=(6,5), side P R is parallel to the x-axis and has a length of 2 . Therefore, triangle P Q R is a right isosceles triangle, where P has measure 90 ° and Q and R each have measure 45°. It follows that if the measure of Q is t °, then t = 45 . Since L=(4,5) and M=(4,7+k), side LM is parallel to the y-axis and has a length of k + 2 . Since L=(4,5) and N=(6+k,5), side LN is parallel to the x-axis and has a length of k + 2 . Therefore, triangle L M N is a right isosceles triangle, where L has measure 90 ° and M and N each have measure 45 °. Of the given choices, only (90-t)° is equal to 45 °, so the measure of N is (90-t)°.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 175 175 of 269 selected Area And Volume M

A triangular prism has a height of 8 centimeters (cm) and a volume of 216 cm3. What is the area, in cm2, of the base of the prism? (The volume of a triangular prism is equal to B h , where B is the area of the base and h is the height of the prism.)

Show Answer Correct Answer: 27

The correct answer is 27 . It's given that a triangular prism has a volume of 216 cubic centimeters (cm3) and the volume of a triangular prism is equal to B h , where B is the area of the base and h is the height of the prism. Therefore, 216=Bh. It's also given that the triangular prism has a height of 8 cm. Therefore, h=8. Substituting 8 for h in the equation 216=Bh yields 216=B(8). Dividing both sides of this equation by 8 yields 27=B. Therefore, the area, in cm2, of the base of the prism is 27 .

Question 176 176 of 269 selected Lines, Angles, And Triangles E
The figure presents triangle A B C, where side A C is horizontal, vertex A is to the left of vertex C, vertex B is above side A C, and the measure of angle A B C is labeled x degrees. Side A C extends horizontally to the right to point D, and the angle, B C D, that is above line segment C D and to the right of side B C has a measure of one hundred ten degrees.

In the given figure, side A C extends to point D. If the measure of angle B A C is equal to the measure of angle B C A, what is the value of x ?

  1. 110

  2. 70

  3. 55

  4. 40

Show Answer Correct Answer: D

Choice D is correct. Since angle B C D and angle B C A form a linear pair of angles, their measures sum to 180°. It’s given that the measure of angle B C D is 110°. Therefore, 110 degrees plus angle B C A, equals 180 degrees. Subtracting 110° from both sides of this equation gives the measure of angle B C A as 70°. It’s also given that the measure of angle B A C is equal to the measure of angle B C A. Thus, the measure of angle B A C is also 70°. The measures of the interior angles of a triangle sum to 180°. Thus, 70 degrees, plus 70 degrees, plus x degrees, equals 180 degrees. Combining like terms on the left-hand side of this equation yields 140 degrees plus x degrees, equals 180 degrees. Subtracting 140° from both sides of this equation yields x degrees equals 40 degrees, or x equals 40.

Choice A is incorrect. This is the value of the measure of angle B C D. Choice B is incorrect. This is the value of the measure of each of the other two interior angles, angle B C A and angle B A C. Choice C is incorrect and may result from an error made when identifying the relationship between the exterior angle of a triangle and the interior angles of the triangle.

Question 177 177 of 269 selected Circles H

A circle has center G , and points M and N lie on the circle. Line segments MH and NH are tangent to the circle at points M and N , respectively. If the radius of the circle is 168 millimeters and the perimeter of quadrilateral GMHN is 3,856 millimeters, what is the distance, in millimeters, between points G and H ?

  1. 168

  2. 1,752

  3. 1,760

  4. 1,768

Show Answer Correct Answer: D

Choice D is correct. It's given that the radius of the circle is 168 millimeters. Since points M and N both lie on the circle, segments G M and G N are both radii. Therefore, segments GM and GN each have length 168 millimeters. Two segments that are tangent to a circle and have a common exterior endpoint have equal length. Therefore, segment MH and segment NH have equal length. Let x represent the length of segment MH. Then x also represents the length of segment NH. It's given that the perimeter of quadrilateral GMHN is 3,856 millimeters. Since the perimeter of a quadrilateral is equal to the sum of the lengths of the sides of the quadrilateral, 3,856=168+168+x+x, or 3,856=336+2x. Subtracting 336 from both sides of this equation yields 3,520 = 2 x , and dividing both sides of this equation by 2 yields 1,760 = x . Therefore, the length of segment MH is 1,760 millimeters. A line segment that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. Therefore, segment GM is perpendicular to segment MH. Since perpendicular segments form right angles, angle GMH is a right angle. Therefore, triangle GMH is a right triangle with legs of length 1,760 millimeters and 168 millimeters, and hypotenuse GH. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b , then a 2 + b 2 = c 2 . Substituting 1,760 for a and 168 for b in this equation yields 1,7602+1682=c2, or 3,125,824 = c 2 . Taking the square root of both sides of this equation yields ±1,768=c. Since c represents a length, which must be positive, the value of c is 1,768 . Therefore, the length of segment GH is 1,768 millimeters, so the distance between points G and H is 1,768 millimeters.

Choice A is incorrect. This is the distance between points G and M and between points G and N , not the distance between points G and H .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the distance between points M and H and between points N and H , not the distance between points G and H .

Question 178 178 of 269 selected Lines, Angles, And Triangles M

Triangles A B C and D E F are congruent, where A corresponds to D , and B and E are right angles. The measure of angle A is 69°. What is the measure, in degrees, of angle F ?

Show Answer Correct Answer: 21

The correct answer is 21. It's given that triangles ABC and DEF are congruent with angle A corresponding to angle D. Corresponding angles of congruent triangles are congruent and, therefore, have equal measure. It's given that the measure of angle A is 69°. It follows that the measure of angle D is also 69°. It's given that angle E is a right angle. Therefore, the measure of angle E is 90°. Let x represent the measure, in degrees, of angle F. Since the measures of the angles in a triangle sum to 180°, it follows that 69+90+x=180, or 159+x=180. Subtracting 159 from both sides of this equation yields x=21. Therefore, the measure, in degrees, of angle F is 21.

Question 179 179 of 269 selected Area And Volume E

Each base of a right rectangular prism has a length of 19 inches and a width of 8 inches. The prism has a volume of 2,736 cubic inches. What is the height, in inches, of the prism?

  1. 18

  2. 27

  3. 144

  4. 152

Show Answer Correct Answer: A

Choice A is correct. The volume, V, of a rectangular prism is given by the formula V=lwh, where l is the length of the base, w is the width of the base, and h is the height of the prism. It’s given that each base of a right rectangular prism has a length of 19 inches and a width of 8 inches, and that the prism has a volume of 2,736 cubic inches. Substituting 19 for l, 8 for w, and 2,736 for V in the formula V=lwh gives 2,736=(19)(8)(h), or 2,736=152h. Dividing each side of this equation by 152 yields 18=h. Therefore, the height, in inches, of the prism is 18.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the area, in square inches, of the base of the prism, not the height, in inches, of the prism.

Question 180 180 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled t, j, and k.
  • Line t intersects both line j and line k.
  • At the intersection of line t and line j, 2 angles are labeled clockwise from top left as follows:
    • Bottom right: 133°
    • Bottom left: x°
  • A note indicates the figure is not drawn to scale.

In the figure, line j is parallel to line k . What is the value of x ?

Show Answer Correct Answer: 47

The correct answer is 47 . Based on the figure, the angle with measure x° and the angle with measure 133° together form a straight line. Therefore, these two angles are supplementary, so the sum of their measures is 180°. It follows that x + 133 = 180 . Subtracting 133 from both sides of this equation yields x = 47 .

Question 181 181 of 269 selected Area And Volume E

  • The lengths of the sides are x, y, and z.
  • A note indicates the figure is not drawn to scale.

 

The triangle shown has a perimeter of 22 units. If x=9 units and y=7 units, what is the value of z , in units?

  1. 6

  2. 7

  3. 9

  4. 16

Show Answer Correct Answer: A

Choice A is correct. The perimeter of a triangle is the sum of the lengths of its three sides. The triangle shown has side lengths x , y , and z . It's given that the triangle has a perimeter of 22 units. Therefore, x + y + z = 22 . If x = 9 units and y = 7 units, the value of z , in units, can be found by substituting 9 for x and 7 for y in the equation x + y + z = 22 , which yields 9+7+z=22, or 16+z=22. Subtracting 16 from both sides of this equation yields z = 6 . Therefore, if x = 9 units and y = 7 units, the value of z , in units, is 6 .

Choice B is incorrect. This is the value of y , in units, not the value of z , in units.

Choice C is incorrect. This is the value of x , in units, not the value of z , in units.

Choice D is incorrect. This is the value of x + y , in units, not the value of z , in units.

Question 182 182 of 269 selected Lines, Angles, And Triangles H

In triangle RST, angle T is a right angle, point L lies on RS¯, point K lies on ST¯, and LK¯ is parallel to RT¯. If the length of RT¯ is 72 units, the length of LK¯ is 24 units, and the area of triangle RST is 792 square units, what is the length of KT¯, in units?

Show Answer Correct Answer: 14.66, 14.67, 44/3

The correct answer is 44 3 . It's given that in triangle R S T , angle T is a right angle. The area of a right triangle can be found using the formula A=12l1l2, where A represents the area of the right triangle, l1 represents the length of one leg of the triangle, and l2 represents the length of the other leg of the triangle. In triangle R S T , the two legs are RT¯ and ST¯. Therefore, if the length of RT¯ is 72 and the area of triangle RST is 792 , then 792=12(72)(ST), or 792=(36)(ST). Dividing both sides of this equation by 36 yields 22=ST. Therefore, the length of ST¯ is 22 . It's also given that point L lies on RS¯, point K lies on ST¯, and LK¯ is parallel to RT¯. It follows that angle LKS is a right angle. Since triangles RST and LSK share angle S and have right angles T and K , respectively, triangles RST and LSK are similar triangles. Therefore, the ratio of the length of RT¯ to the length of LK¯ is equal to the ratio of the length of ST¯ to the length of SK¯. If the length of RT¯ is 72 and the length of LK¯ is 24 , it follows that the ratio of the length of RT¯ to the length of LK¯ is 7224, or 3 , so the ratio of the length of ST¯ to the length of SK¯ is 3 . Therefore, 22SK=3. Multiplying both sides of this equation by SK yields 22=(3)(SK). Dividing both sides of this equation by 3 yields 223=SK. Since the length of ST¯, 22 , is the sum of the length of SK¯, 22 3 , and the length of KT¯, it follows that the length of KT¯ is 22-223, or 44 3 . Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.

Question 183 183 of 269 selected Circles H

A circle has center O , and points A and B lie on the circle. The measure of arc A B is 45 and the length of arc A B is 3 inches. What is the circumference, in inches, of the circle?

  1. 3

  2. 6

  3. 9

  4. 24

Show Answer Correct Answer: D

Choice D is correct. It’s given that the measure of arc AB is 45° and the length of arc AB is 3 inches. The arc measure of the full circle is 360°. If x represents the circumference, in inches, of the circle, it follows that 45°360°=3 inchesx inches. This equation is equivalent to 45360=3x, or 18=3x. Multiplying both sides of this equation by 8x yields 1(x)=3(8), or x=24. Therefore, the circumference of the circle is 24 inches.

Choice A is incorrect. This is the length of arc AB.

Choice B is incorrect and may result from multiplying the length of arc AB by 2 .

Choice C is incorrect and may result from squaring the length of arc AB.

Question 184 184 of 269 selected Circles M

In the xy-plane, a circle with radius 5 has center with coordinates negative 8 comma 6. Which of the following is an equation of the circle?

  1. open parenthesis, x minus 8, close parenthesis, squared, plus, open parenthesis, y plus 6, close parenthesis, squared, equals 25

  2. open parenthesis, x plus 8, close parenthesis, squared, plus, open parenthesis, y minus 6, close parenthesis, squared, equals 25

  3. open parenthesis, x minus 8, close parenthesis, squared, plus, open parenthesis, y plus 6, close parenthesis, squared, equals 5

  4. open parenthesis, x plus 8, close parenthesis, squared, plus, open parenthesis, y minus 6, close parenthesis, squared, equals 5

Show Answer Correct Answer: B

Choice B is correct. An equation of a circle is open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared, where the center of the circle is the point with coordinates h comma k and the radius is r. It’s given that the center of this circle is the point with coordinates negative 8 comma 6 and the radius is 5. Substituting these values into the equation gives open parenthesis, x minus negative 8, close parenthesis, squared, plus, open parenthesis, y minus 6, close parenthesis, squared, equals 5 squared, or open parenthesis, x plus 8, close parenthesis, squared, plus, open parenthesis, y minus 6, close parenthesis, squared, equals 25.

Choice A is incorrect. This is an equation of a circle that has center 8 comma negative 6. Choice C is incorrect. This is an equation of a circle that has center 8 comma negative 6 and radius the square root of 5. Choice D is incorrect. This is an equation of a circle that has radius  the square root of 5.

 

Question 185 185 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled, c, s, and t.
  • Line c intersects both line s and line t.
  • At the intersection of line c and line s, 1 angle is labeled clockwise from top left as follows:
    • Top right: x°
  • At the intersection of line c and line t, 1 angle is labeled clockwise from top left as follows:
    • Top left: 110°
  • A note indicates the figure is not drawn to scale.

In the figure shown, line c intersects parallel lines s and t . What is the value of x ?

Show Answer Correct Answer: 70

The correct answer is 70 . Based on the figure, the angle with measure 110° and the angle vertical to the angle with measure x° are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure x° also has measure x°. It’s given that lines s and t are parallel. Therefore, same side interior angles between lines s and t are supplementary. It follows that x + 110 = 180 . Subtracting 110 from both sides of this equation yields x = 70 .

Question 186 186 of 269 selected Lines, Angles, And Triangles E

Triangles A B C and D E F are congruent, where A corresponds to D , and B and E are right angles. The measure of angle A is 18°. What is the measure of angle F ?

  1. 18 °

  2. 72 °

  3. 90 °

  4. 162 °

Show Answer Correct Answer: B

Choice B is correct. It’s given that triangle A B C is congruent to triangle D E F . Corresponding angles of congruent triangles are congruent and, therefore, have equal measure. It’s given that angle A corresponds to angle D , and that the measure of angle A is 18°. It's also given that the measures of angles B and E are 90°. Since these angles have equal measure, they are corresponding angles. It follows that angle C corresponds to angle F . Let x° represent the measure of angle C . Since the sum of the measures of the interior angles of a triangle is 180°, it follows that 18°+90°+x°=180°, or 108°+x°=180°. Subtracting 108° from both sides of this equation yields x°=72°. Therefore, the measure of angle C is 72°. Since angle C corresponds to angle F , it follows that the measure of angle F is also 72°.

Choice A is incorrect. This is the measure of angle D , not the measure of angle F .

Choice C is incorrect. This is the measure of angle E , not the measure of angle F .

Choice D is incorrect. This is the sum of the measures of angles E and F , not the measure of angle F .

Question 187 187 of 269 selected Area And Volume H

A right circular cone has a volume of 71,148 π cubic centimeters and the area of its base is 5,929 π square centimeters. What is the slant height, in centimeters, of this cone?

  1. 12

  2. 36

  3. 77

  4. 85

Show Answer Correct Answer: D

Choice D is correct. The volume, V , of a right circular cone is given by the formula V=13πr2h, where π r 2 is the area of the circular base of the cone and h is the height. It’s given that this right circular cone has a volume of 71,148 π cubic centimeters and the area of its base is 5,929 π square centimeters. Substituting 71,148 π for V and 5,929 π for π r 2 in the formula V=13πr2h yields 71,148π=(13)(5,929π)(h). Dividing each side of this equation by 5,929 π yields 12 = h 3 . Multiplying each side of this equation by 3 yields 36 = h . Let s represent the slant height, in centimeters, of this cone. A right triangle is formed by the radius, r , height, h , and slant height, s , of this cone, where r and h are the legs of the triangle and s is the hypotenuse. Using the Pythagorean theorem, the equation r2+h2=s2 represents this relationship. Because 5,929 π is the area of the base and the area of the base is π r 2 , it follows that 5,929 π = π r 2 . Dividing both sides of this equation by π yields 5,929 = r 2 . Substituting 5,929 for r 2 and 36 for h in the equation r2+h2=s2 yields 5,929+362=s2, which is equivalent to 5,929+1,296=s2, or 7,225 = s 2 . Taking the positive square root of both sides of this equation yields 85 = s . Therefore, the slant height of the cone is 85 centimeters.

Choice A is incorrect. This is one-third of the height, in centimeters, not the slant height, in centimeters, of this cone.

Choice B is incorrect. This is the height, in centimeters, not the slant height, in centimeters, of this cone.

Choice C is incorrect. This is the radius, in centimeters, of the base, not the slant height, in centimeters, of this cone.

Question 188 188 of 269 selected Area And Volume M

A triangle has a base length of 10 centimeters and a corresponding height of 70 centimeters. What is the area, in square centimeters, of the triangle?

  1. 700

  2. 350

  3. 175

  4. 80

Show Answer Correct Answer: B

Choice B is correct. The area, A, of a triangle is given by A=(12)bh, where b is the length of a base of the triangle and h is the corresponding height of the triangle. It's given that a triangle has a base length of 10 centimeters and a corresponding height of 70 centimeters. Substituting 10 for b and 70 for h in the formula A=(12)bh yields A=(12)(10)(70), or A=350. Therefore, the area, in square centimeters, of the triangle is 350.

Choice A is incorrect. This is the product of the given base and height of the triangle, not its area. 

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the sum of the given base and height of the triangle, not its area. 

Question 189 189 of 269 selected Circles H

Which of the following equations represents a circle in the xy-plane that intersects the y-axis at exactly one point?

  1. ( x - 8 ) 2 + ( y - 8 ) 2 = 16

  2. ( x - 8 ) 2 + ( y - 4 ) 2 = 16

  3. ( x - 4 ) 2 + ( y - 9 ) 2 = 16

  4. x 2 + ( y - 9 ) 2 = 16

Show Answer Correct Answer: C

Choice C is correct. The graph of the equation (x-h)2+(y-k)2=r2 in the xy-plane is a circle with center (h,k) and a radius of length r . The radius of a circle is the distance from the center of the circle to any point on the circle. If a circle in the xy-plane intersects the y-axis at exactly one point, then the perpendicular distance from the center of the circle to this point on the y-axis must be equal to the length of the circle's radius. It follows that the x-coordinate of the circle's center must be equivalent to the length of the circle's radius. In other words, if the graph of (x-h)2+(y-k)2=r2 is a circle that intersects the y-axis at exactly one point, then r=|h| must be true. The equation in choice C is (x-4)2+(y-9)2=16, or (x-4)2+(y-9)2=42. This equation is in the form (x-h)2+(y-k)2=r2, where h=4, k=9, and r=4, and represents a circle in the xy-plane with center (4,9) and radius of length 4 . Substituting 4 for r and 4 for h in the equation r=|h| yields 4=|4|, or 4=4, which is true. Therefore, the equation in choice C represents a circle in the xy-plane that intersects the y-axis at exactly one point. 

Choice A is incorrect. This is the equation of a circle that does not intersect the y-axis at any point.

Choice B is incorrect. This is an equation of a circle that intersects the x-axis, not the y-axis, at exactly one point.

Choice D is incorrect. This is the equation of a circle with the center located on the y-axis and thus intersects the y-axis at exactly two points, not exactly one point.

Question 190 190 of 269 selected Right Triangles And Trigonometry E

A right triangle has legs with lengths of 11 centimeters and 9 centimeters. What is the length of this triangle's hypotenuse, in centimeters?

  1. 40

  2. 202

  3. 20

  4. 202

Show Answer Correct Answer: B

Choice B is correct. The Pythagorean theorem states that for a right triangle, c2=a2+b2, where c represents the length of the hypotenuse and a and b represent the lengths of the legs. It’s given that a right triangle has legs with lengths of 11 centimeters and 9 centimeters. Substituting 11 for a and 9 for b in the formula c2=a2+b2 yields c2=112+92, which is equivalent to c2=121+81, or c2=202. Taking the square root of each side of this equation yields c=±202. Since c represents a length, c must be positive. Therefore, the length of the triangle’s hypotenuse, in centimeters, is 202.

Choice A is incorrect. This is the result of solving the equation c2=11(2)+9(2), not c2=112+92.

Choice C is incorrect. This is the result of solving the equation c(2)=11(2)+9(2), not c2=112+92.

Choice D is incorrect. This is the result of solving the equation c=112+92, not c2=112+92.

Question 191 191 of 269 selected Right Triangles And Trigonometry M
The figure presents triangle A, B C, where side A, C is horizontal and point B is above side A, C. Point D lies on side A, C, and line segment B D is drawn, forming right angle B D C. Side B C is labeled 12. Angle A, B D is labeled 30 degrees and angle B C D is labeled 60 degrees.

In triangle A, B C above, what is the length of  segment A, D?

  1. 4

  2. 6

  3. 6 times the square root of 2

  4. 6 times the square root of 3

Show Answer Correct Answer: B

Choice B is correct. Triangles ADB and CDB are both 30 degree 60 degree 90 degree triangles and share side B D . Therefore, triangles ADB and CDB are congruent by the angle-side-angle postulate. Using the properties of 30 degree 60 degree 90 degree triangles, the length of side A, D is half the length of hypotenuse A, B . Since the triangles are congruent, the length of side A, B equals the length of side B C, which equals 12. So the length of side A, D is twelve halves equals 6.

Alternate approach: Since angle CBD has a measure of 30 degrees, angle ABC must have a measure of 60 degrees. It follows that triangle ABC is equilateral, so side AC also has length 12. It also follows that the altitude BD is also a median, and therefore the length of AD is half of the length of AC, which is 6.

Choice A is incorrect. If the length of side A, D were 4, then the length of side A, B would be 8. However, this is incorrect because side A, B is congruent to side B C, which has a length of 12. Choices C and D are also incorrect. Following the same procedures as used to test choice A gives side A, B a length of 12 times the square root of 2 for choice C and 12 times the square root of 3 for choice D. However, these results cannot be true because side A, B is congruent to side B C, which has a length of 12.

 

Question 192 192 of 269 selected Area And Volume E

What is the area, in square centimeters, of a rectangle with a length of 36 centimeters and a width of 34 centimeters?

  1. 70

  2. 140

  3. 1,156

  4. 1,224

Show Answer Correct Answer: D

Choice D is correct. The area A , in square centimeters, of a rectangle can be found using the formula A=lw, where l is the length, in centimeters, of the rectangle and w is its width, in centimeters. It's given that the rectangle has a length of 36 centimeters and a width of 34 centimeters. Substituting 36 for l and 34 for w in the formula A=lw yields A=36(34), or A=1,224. Therefore, the area, in square centimeters, of this rectangle is 1,224.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the perimeter, in centimeters, not the area, in square centimeters, of the rectangle.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 193 193 of 269 selected Area And Volume M

The length of each side of a square is 94 centimeters (cm). Which expression gives the area, in cm2, of the square?

  1. 2·94

  2. 2·94·94

  3. 4·94

  4. 94·94

Show Answer Correct Answer: D

Choice D is correct. The area of a square is given by s2, where s is the length of each side of the square. It's given that the length of each side of a square is 94 cm. It follows that the area, in cm2, of the square is (94)2, or 94·94. Therefore, the expression that gives the area, in cm2, of the square is 94·94.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect. This expression gives the perimeter, in cm, of the square.

Question 194 194 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled k, m, and n.
  • Line k intersects both line m and line n.
  • At the intersection of line k and line n, 2 angles are labeled clockwise from top left as follows:
    • Top left: x°
    • Bottom right: 145°
  • A note indicates the figure is not drawn to scale.

In the figure, line m is parallel to line n , and line k intersects both lines. Which of the following statements is true?

  1. The value of x is less than 145 .

  2. The value of x is greater than 145 .

  3. The value of x is equal to 145 .

  4. The value of x cannot be determined.

Show Answer Correct Answer: C

Choice C is correct. Vertical angles, or angles that are opposite each other when two lines intersect, are congruent. It’s given that line k intersects line n . Based on the figure, the angle with measure x° and the angle with measure 145° are vertical angles. Therefore, the value of x is equal to 145 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 195 195 of 269 selected Circles H

Open parenthesis, x minus 6, close parenthesis, squared, plus, open parenthesis, y plus 5, close parenthesis, squared, equals 16

In the xy-plane, the graph of the equation above is a circle. Point P is on the circle and has coordinates 10 comma negative 5. If line segment P Q is a diameter of the circle, what are the coordinates of point Q ?

  1. 2 comma negative 5

  2. 6 comma negative 1

  3. 6 comma negative 5

  4. 6 comma negative 9

Show Answer Correct Answer: A

Choice A is correct. The standard form for the equation of a circle is open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared, where the ordered pair h comma k are the coordinates of the center and r is the length of the radius. According to the given equation, the center of the circle is the point with coordinates 6 comma negative 5. Let x sub 1 comma y sub 1 represent the coordinates of point Q. Since point P 10 comma negative 5 and point Q x sub 1 comma y sub 1 are the endpoints of a diameter of the circle, the center with coordinates 6 comma negative 5 lies on the diameter, halfway between P and Q. Therefore, the following relationships hold: the fraction with numerator x sub 1, plus 10, and denominator 2, equals 6 and the fraction with numerator y sub 1, plus negative 5, and denominator 2, equals negative 5. Solving the equations for x sub 1 and y sub 1, respectively, yields x sub 1, equals 2 and y sub 1, equals negative 5. Therefore, the coordinates of point Q are 2 comma negative 5.

Alternate approach: Since point P 10 comma negative 5 on the circle and the center of the circle 6 comma negative 5 have the same y-coordinate, it follows that the radius of the circle is 10 minus 6, equals 4. In addition, the opposite end of the diameter P Q must have the same y-coordinate as P and be 4 units away from the center. Hence, the coordinates of point Q must be 2 comma negative 5.

Choices B and D are incorrect because the points given in these choices lie on a diameter that is perpendicular to the diameter P Q. If either of these points were point Q, then line segment P Q would not be the diameter of the circle. Choice C is incorrect because the point with coordinates 6 comma negative 5 is the center of the circle and does not lie on the circle.

 

Question 196 196 of 269 selected Circles H

A circle in the xy-plane has its center at (-5,2) and has a radius of 9 . An equation of this circle is x2+y2+ax+by+c=0, where a , b , and c are constants. What is the value of c ?

Show Answer Correct Answer: -52

The correct answer is -52 . The equation of a circle in the xy-plane with its center at (h,k) and a radius of r can be written in the form (x-h)2+(y-k)2=r2. It's given that a circle in the xy-plane has its center at (-5,2) and has a radius of 9 . Substituting -5 for h , 2 for k , and 9 for r in the equation (x-h)2+(y-k)2=r2 yields (x-(-5))2+(y-2)2=92, or (x+5)2+(y-2)2=81. It's also given that an equation of this circle is x2+y2+ax+by+c=0, where a , b , and c are constants. Therefore, (x+5)2+(y-2)2=81 can be rewritten in the form x2+y2+ax+by+c=0. The equation (x+5)2+(y-2)2=81, or (x+5)(x+5)+(y-2)(y-2)=81, can be rewritten as x2+5x+5x+25+y2-2y-2y+4=81. Combining like terms on the left-hand side of this equation yields x2+y2+10x-4y+29=81. Subtracting 81 from both sides of this equation yields x2+y2+10x-4y-52=0, which is equivalent to x2+y2+10x+(-4)y+(-52)=0. This equation is in the form x2+y2+ax+by+c=0. Therefore, the value of c is -52 .

Question 197 197 of 269 selected Circles H

In the xy-plane, a circle has center C with coordinates (h,k). Points A and B lie on the circle. Point A has coordinates (h+1,k+102), and ACB is a right angle. What is the length of AB¯?

  1. 206

  2. 2 102

  3. 103 2

  4. 103 3

Show Answer Correct Answer: A

Choice A is correct. It's given that points A and B lie on the circle with center C . Therefore, AC¯ and BC¯ are both radii of the circle. Since all radii of a circle are congruent, AC¯ is congruent to BC¯. The length of AC¯, or the distance from point A to point C , can be found using the distance formula, which gives the distance between two points, (x1,y1) and (x2,y2), as (x1-x2)2+(y1-y2)2. Substituting the given coordinates of point A , (h+1,k+102), for (x1,y1) and the given coordinates of point C , (h,k), for (x2,y2) in the distance formula yields (h+1-h)2+(k+102-k)2, or 12+(102)2, which is equivalent to 1+102, or 103 . Therefore, the length of AC¯ is 103 and the length of BC¯ is 103 . It's given that angle ACB is a right angle. Therefore, triangle ACB is a right triangle with legs AC¯ and BC¯ and hypotenuse AB¯. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b , then a 2 + b 2 = c 2 . Substituting 103 for a and b in this equation yields (103)2+(103)2=c2, or 103+103=c2, which is equivalent to 206 = c 2 . Taking the positive square root of both sides of this equation yields 206 = c . Therefore, the length of AB¯ is 206 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This would be the length of AB¯ if the length of AC¯ were 103 , not 103 .

Choice D is incorrect and may result from conceptual or calculation errors.

Question 198 198 of 269 selected Circles E

( x - 6 ) 2 + ( y - 3 ) 2 = 81

The graph of the given equation in the xy-plane is a circle. What is the length of the radius of this circle?

  1. 3

  2. 6

  3. 9

  4. 81

Show Answer Correct Answer: C

Choice C is correct. The equation of a circle in the xy-plane can be written as (x-h)2+(y-k)2=r2, where the center of the circle is (h,k) and the radius of the circle is r. The graph of the given equation, (x-6)2+(y-3)2=81, is a circle in the xy-plane. This equation can be written as (x-6)2+(y-3)2=92, where h=6, k=3, and r=9. Therefore, the radius of this circle is 9.

Choice A is incorrect. This is the y-coordinate of the center, not the radius, of the circle defined by the given equation.

Choice B is incorrect. This is the x-coordinate of the center, not the radius, of the circle defined by the given equation.

Choice D is incorrect. This is the value of the radius squared, not the radius, of the circle defined by the given equation.

Question 199 199 of 269 selected Right Triangles And Trigonometry H

The length of a rectangle’s diagonal is 3 17 , and the length of the rectangle’s shorter side is 3 . What is the length of the rectangle’s longer side?

Show Answer Correct Answer: 12

The correct answer is 12 . The diagonal of a rectangle forms a right triangle, where the shorter side and the longer side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. It's given that the length of the rectangle's diagonal is 317 and the length of the rectangle's shorter side is 3 . Thus, the length of the hypotenuse of the right triangle formed by the diagonal is 317 and the length of one of the legs is 3 . By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b , then a2+b2=c2. Substituting 317 for c and 3 for b in this equation yields a2+(3)2=(317)2, or a2+9=153. Subtracting 9 from both sides of this equation yields a2=144. Taking the square root of both sides of this equation yields a=±144, or a=±12. Since a represents a length, which must be positive, the value of a is 12 . Thus, the length of the rectangle's longer side is 12 .

Question 200 200 of 269 selected Right Triangles And Trigonometry E

  • One angle is a right angle.
  • The length of the side opposite the right angle is c.
  • The lengths of the other 2 sides are as follows:
    • 9
    • 6
  • A note indicates the figure is not drawn to scale.

In the right triangle shown, which of the following is closest to the value of c ?

  1. 7.5

  2. 10.8

  3. 15

  4. 58.5

Show Answer Correct Answer: B

Choice B is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length t and legs with lengths r and s, then r2+s2=t2. It's given in the right triangle shown that the legs have lengths of 9 and 6 and the hypotenuse has a length of c. Substituting 9 for r, 6 for s, and c for t in r2+s2=t2 yields 92+62=c2, or 117=c2. Taking the square root of both sides of this equation yields ±117=c. Since the length of a side of a triangle must be positive, the value of c is 117, which is approximately equal to 10.8167. Of the choices, 10.8 is the closest to the value of c.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 201 201 of 269 selected Circles H

The equation x 2 + ( y - 2 ) 2 = 36 represents circle A. Circle B is obtained by shifting circle A down 4 units in the xy-plane. Which of the following equations represents circle B?

  1. x 2 + ( y + 2 ) 2 = 36

  2. x 2 + ( y - 6 ) 2 = 36

  3. ( x - 4 ) 2 + ( y - 2 ) 2 = 36

  4. ( x + 4 ) 2 + ( y - 2 ) 2 = 36

Show Answer Correct Answer: A

Choice A is correct. The standard form of an equation of a circle in the xy-plane is (x-h)2+(y-k)2=r2, where the coordinates of the center of the circle are (h,k) and the length of the radius of the circle is r . The equation of circle A, x2+(y-2)2=36, can be rewritten as (x-0)2+(y-2)2=62. Therefore, the center of circle A is at (0,2) and the length of the radius of circle A is 6 . If circle A is shifted down 4 units, the y-coordinate of its center will decrease by 4 ; the radius of the circle and the x-coordinate of its center will not change. Therefore, the center of circle B is at (0,2-4), or (0,-2), and its radius is 6 . Substituting 0 for h , -2 for k , and 6 for r in the equation (x-h)2+(y-k)2=r2 yields (x-0)2+(y-(-2))2=(6)2, or x2+(y+2)2=36. Therefore, the equation x2+(y+2)2=36 represents circle B.

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, 4 units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, 4 units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, 4 units.

Question 202 202 of 269 selected Area And Volume H

Two identical rectangular prisms each have a height of 90 centimeters (cm). The base of each prism is a square, and the surface area of each prism is K cm2. If the prisms are glued together along a square base, the resulting prism has a surface area of 9247K cm2. What is the side length, in cm, of each square base?

  1. 4

  2. 8

  3. 9

  4. 16

Show Answer Correct Answer: B

Choice B is correct. Let x represent the side length, in cm, of each square base. If the two prisms are glued together along a square base, the resulting prism has a surface area equal to twice the surface area of one of the prisms, minus the area of the two square bases that are being glued together, which yields 2K-2x2 cm2 . It’s given that this resulting surface area is equal to 9247K cm2, so 2K-2x2=9247K. Subtracting 9247K from both sides of this equation yields 2K-9247K-2x2=0. This equation can be rewritten by multiplying 2 K on the left-hand side by 4747, which yields 9447K-9247K-2x2=0, or 247K-2x2=0. Adding 2x2 to both sides of this equation yields 247K=2x2. Multiplying both sides of this equation by 472 yields K=47x2. The surface area K , in cm2, of each rectangular prism is equivalent to the sum of the areas of the two square bases and the areas of the four lateral faces. Since the height of each rectangular prism is 90 cm and the side length of each square base is x cm, it follows that the area of each square base is x2 cm2 and the area of each lateral face is 90x cm2. Therefore, the surface area of each rectangular prism can be represented by the expression 2x2+4(90x), or 2x2+360x. Substituting this expression for K in the equation K=47x2 yields 2x2+360x=47x2. Subtracting 2x2 and 360 x from both sides of this equation yields 0=45x2-360x. Factoring x from the right-hand side of this equation yields 0=x(45x-360). Applying the zero product property, it follows that x=0 and 45x-360=0. Adding 360 to both sides of the equation 45x-360=0 yields 45x=360. Dividing both sides of this equation by 45 yields x=8. Since a side length of a rectangular prism can’t be 0 , the length of each square base is 8 cm.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 203 203 of 269 selected Circles M
The figure presents a circle in the x y plane, with the origin labeled O. The center of the circle lies at the origin. There are 4 points, T, R, Q, and P, labeled clockwise on the circle. Point T lies on the x axis, directly to the left of the origin, and has coordinates negative 1 comma 0. Point R lies above the x axis, and to the left of the y axis. Point Q lies above the x axis and to the right of the y axis. Point P lies on the x axis, directly to the right of the origin, and has coordinates 1 comma 0. Points R and Q appear to be symmetric with respect to the y axis. Four line segments are drawn from the origin to each of the four points on the circle.

In the xy-plane above, points P, Q, R, and T lie on the circle with center O. The degree measures of angles P O Q and R O T are each 30°. What is the radian measure of angle Q O R ?

  1. five sixths, pi

  2. three fourths, pi

  3. two thirds, pi

  4. one third, pi

Show Answer Correct Answer: C

Choice C is correct. Because points T, O, and P all lie on the x-axis, they form a line. Since the angles on a line add up to 180 degrees, and it’s given that angles POQ and ROT each measure 30 degrees , it follows that the measure of angle QOR is 180 degrees minus 30 degrees, minus 30 degrees, equals 120 degrees. Since the arc of a complete circle is 360 degrees or 2 pi radians, a proportion can be set up to convert the measure of angle QOR from degrees to radians: the fraction 360 degrees over 2 pi radians, equals, the fraction 120 degrees over x radians, where x is the radian measure of angle QOR. Multiplying each side of the proportion by 2 pi x gives 360 x equals 240 pi . Solving for x gives the fraction 240 over 360 times pi, or two thirds pi.

Choice A is incorrect and may result from subtracting only angle POQ from 180 degrees to get a value of 150 degrees and then finding the radian measure equivalent to that value. Choice B is incorrect and may result from a calculation error. Choice D is incorrect and may result from calculating the sum of the angle measures, in radians, of angles POQ and ROT.

 

Question 204 204 of 269 selected Lines, Angles, And Triangles H

 

  • Triangle upper L upper M upper R has a common vertex with triangle upper P upper Q upper R.
  • The common vertex is upper R.
  • A note indicates the figure is not drawn to scale.

 

In the figure, LQ¯ intersects MP¯ at point R , and LM¯ is parallel to PQ¯. The lengths of MR¯LR¯, and RP¯ are 6 , 7 , and 11 , respectively. What is the length of LQ¯?

  1. 119 11

  2. 77 6

  3. 113 6

  4. 119 6

Show Answer Correct Answer: D

Choice D is correct. The figure shows that angle MRL and angle PRQ are vertical angles. Since vertical angles are congruent, angle MRL and angle PRQ are congruent. It’s given that LM¯ is parallel to PQ¯. The figure also shows that LQ¯ intersects LM¯ and PQ¯. If two parallel segments are intersected by a third segment, alternate interior angles are congruent. Thus, alternate interior angles MLR and PQR are congruent. Since triangles LMR and PQR have two pairs of congruent angles, the triangles are similar. Sides LR and MR in triangle LMR correspond to sides RQ and RP, respectively, in triangle PQR. Since the lengths of corresponding sides in similar triangles are proportional, it follows that RQLR=RPMR. It's given that the lengths of MR¯, LR¯, and RP¯ are 6 , 7 , and 11 , respectively. Substituting 6 for MR, 7 for LR, and 11 for RP in the equation RQLR=RPMR yields RQ7=116. Multiplying each side of this equation by 7 yields RQ=(116)(7), or RQ=776. It's given that LQ¯ intersects MP¯ at point R , so LQ=LR+RQ. Substituting 7 for LR and 776 for RQ in this equation yields LQ=7+776, or LQ=1196. Therefore, the length of LQ¯ is 1196.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the length of RQ¯, not LQ¯.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 205 205 of 269 selected Lines, Angles, And Triangles M
The figure presents triangle A C E, where side A E is horizontal and vertex E is to the right of vertex A. Vertex C lies above side A E. Point B lies on side A C, point D lies on side C E, and line segment B D is drawn. A note states that the figure is not drawn to scale.

In the figure above, segments AE and BD are parallel. If angle BDC measures 58° and angle ACE measures 62°, what is the measure of angle CAE ?

  1. 58°

  2. 60°

  3. 62°

  4. 120°

Show Answer Correct Answer: B

Choice B is correct. It’s given that angle ACE measures 62 degrees. Since segments AE and BD are parallel, angles BDC and CEA are congruent. Therefore, angle CEA measures 58 degrees. The sum of the measures of angles ACE, CEA, and CAE is 180 degrees since the sum of the interior angles of triangle ACE is equal to 180 degrees . Let the measure of angle CAE be x degrees. Therefore, 62 plus 58, plus x, equals 180, which simplifies to x equals 60. Thus, the measure of angle CAE is 60 degrees.

Choice A is incorrect. This is the measure of angle AEC, not that of angle CAE. Choice C is incorrect. This is the measure of angle ACE, not that of CAE. Choice D is incorrect. This is the sum of the measures of angles ACE and CEA.

Question 206 206 of 269 selected Circles M

  • The center of the circle is point upper O.
  • Points upper S, upper R, upper Q, and upper P are on the circle.
  • Line segment upper P upper R is a diameter of the circle.
  • Line segment upper Q upper S is a diameter of the circle.
  • Diameters upper P upper R and upper Q upper S intersect at point upper O.
  • A note indicates the figure is not drawn to scale.

The circle shown has center O , circumference 144π, and diameters PR¯ and QS¯. The length of arc P S is twice the length of arc P Q . What is the length of arc Q R ?

  1. 24π

  2. 48π

  3. 72π

  4. 96π

Show Answer Correct Answer: B

Choice B is correct. Since PR¯ and QS¯ are diameters of the circle shown, OS¯OR¯, OP¯, and OQ¯ are radii of the circle and are therefore congruent. Since SOP and ROQ are vertical angles, they are congruent. Therefore, arc PS and arc QR are formed by congruent radii and have the same angle measure, so they are congruent arcs. Similarly, SOR and POQ are vertical angles, so they are congruent. Therefore, arc SR and arc PQ are formed by congruent radii and have the same angle measure, so they are congruent arcs. Let x represent the length of arc SR. Since arc SR and arc PQ are congruent arcs, the length of arc PQ can also be represented by x . It’s given that the length of arc PS is twice the length of arc PQ. Therefore, the length of arc PS can be represented by the expression 2x. Since arc PS and arc QR are congruent arcs, the length of arc QR can also be represented by 2x. This gives the expression x+x+2x+2x. Since it's given that the circumference is 144π, the expression x+x+2x+2x is equal to 144π. Thus x+x+2x+2x=144π, or 6x=144π. Dividing both sides of this equation by 6 yields x=24π. Therefore, the length of arc QR is 2(24π), or 48π.

Choice A is incorrect. This is the length of arc PQ, not arc QR.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 207 207 of 269 selected Lines, Angles, And Triangles H

In convex pentagon A B C D E , segment A B is parallel to segment D E . The measure of angle B is 139 degrees, and the measure of angle D is 174 degrees. What is the measure, in degrees, of angle C ?

Show Answer Correct Answer: 47

The correct answer is 47 . It's given that the measure of angle B is 139 degrees. Therefore, the exterior angle formed by extending segment A B at point B has measure 180-139, or 41 , degrees. It's given that segment A B is parallel to segment D E . Extending segment B C at point C and extending segment D E at point D until the two segments intersect results in a transversal that intersects two parallel line segments. One of these intersection points is point B , and let the other intersection point be point X . Since segment A B is parallel to segment D E , alternate interior angles are congruent. Angle CXD and the exterior angle formed by extending segment A B at point B are alternate interior angles. Therefore, the measure of angle CXD is 41 degrees. It's given that the measure of angle D in pentagon A B C D E is 174 degrees. Therefore, angle C D X has measure 180-174, or 6 , degrees. Since angle C in pentagon A B C D E is an exterior angle of triangle C D X , it follows that the measure of angle C is the sum of the measures of angles C D X and CXD. Therefore, the measure, in degrees, of angle C is 6+41, or 47 .

Alternate approach: A line can be created that's perpendicular to segments A B and D E and passes through point C . Extending segments A B and D E at points B and D , respectively, until they intersect this line yields two right triangles. Let these intersection points be point X and point Y , and the two right triangles be triangle BXC and triangle DYC. It's given that the measure of angle B is 139 degrees. Therefore, angle CBX has measure 180-139, or 41 , degrees. Since the measure of angle CBX is 41 degrees and the measure of angle BXC is 90 degrees, it follows that the measure of angle XCB is 180-90-41, or 49 , degrees. It's given that the measure of angle D is 174 degrees. Therefore, angle YDC has measure 180-174, or 6 , degrees. Since the measure of angle YDC is 6 degrees and the measure of angle CYD is 90 degrees, it follows that the measure of angle DCY is 180-90-6, or 84 , degrees. Since angles XCBDCY, and angle C in pentagon A B C D E form segment X Y , it follows that the sum of the measures of those angles is 180 degrees. Therefore, the measure, in degrees, of angle C is 180-49-84, or 47 .

Question 208 208 of 269 selected Area And Volume H

The volume of right circular cylinder A is 22 cubic centimeters. What is the volume, in cubic centimeters, of a right circular cylinder with twice the radius and half the height of cylinder A?

  1. 11

  2. 22

  3. 44

  4. 66

Show Answer Correct Answer: C

Choice C is correct. The volume of right circular cylinder A is given by the expression pi r squared times h, where r is the radius of its circular base and h is its height. The volume of a cylinder with twice the radius and half the height of cylinder A is given by pi times, open parenthesis, 2 r, close parenthesis, squared, times one half h, which is equivalent to 4 pi r squared, times, one half h, and equals 2 pi r squared times h. Therefore, the volume is twice the volume of cylinder A, or 2 times 22, equals 44.

Choice A is incorrect and likely results from not multiplying the radius of cylinder A by 2. Choice B is incorrect and likely results from not squaring the 2 in 2r when applying the volume formula. Choice D is incorrect and likely results from a conceptual error.

 

Question 209 209 of 269 selected Right Triangles And Trigonometry H

Which of the following expressions is equivalent to (sin24°)(cos66°)+(cos24°)(sin66°)?

  1. 2(cos66°)(sin24°)

  2. 2(cos66°)+2(cos24°)

  3. (cos66°)2+(cos24°)2

  4. (cos66°)2+(sin24°)2

Show Answer Correct Answer: C

Choice C is correct. The sine of an angle is equal to the cosine of its complementary angle. Since angles with measures 24° and 66° are complementary to each other, sin24° is equal to cos66° and sin66° is equal to cos24°. Substituting cos66° for sin24° and cos24° for sin66° in the given expression yields (cos66°)(cos66°)+(cos24°)(cos24°), or (cos66°)2+(cos24°)2.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 210 210 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled n, r, and s.
  • Line n intersects both line r and line s.
  • At the intersection of line n and line r, 1 angle is labeled clockwise from top left as follows:
    • Top left: 162°
  • At the intersection of line n and line s, 1 angle is labeled clockwise from top left as follows:
    • Top left: x°
  • A note indicates the figure is not drawn to scale.

In the figure, line n intersects lines r and s . Line r is parallel to line s . What is the value of x ?

Show Answer Correct Answer: 162

The correct answer is 162 . It's given that line r is parallel to line s . Since line n intersects both lines r and s , it's a transversal. The angles in the figure marked as 162° and x° are on the same side of the transversal, where one is an interior angle with line s as a side, and the other is an exterior angle with line r as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that the value of x is 162 .

Question 211 211 of 269 selected Right Triangles And Trigonometry E

  • The side lengths are labeled as follows:
    • a
    • b
    • c
  • The angle opposite the side labeled c is a right angle.
  • A note indicates the figure is not drawn to scale.

For the right triangle shown, a=4 and b=5. Which expression represents the value of c ?

  1. 4+5

  2. (4)(5)

  3. 4+5

  4. 42+52

Show Answer Correct Answer: D

Choice D is correct. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b , then c2=a2+b2. In the right triangle shown, the hypotenuse has length c and the legs have lengths a and b . It's given that a=4 and b=5. Substituting 4 for a and 5 for b in the Pythagorean theorem yields c2=42+52. Taking the square root of both sides of this equation yields c=±42+52. Since the length of a side of a triangle must be positive, the value of c is 42+52.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 212 212 of 269 selected Area And Volume E

The area of a rectangle is 630 square inches. The length of the rectangle is 70 inches. What is the width, in inches, of this rectangle?

  1. 9

  2. 70

  3. 315

  4. 560

Show Answer Correct Answer: A

Choice A is correct. The area A , in square inches, of a rectangle is the product of its length l, in inches, and its width w , in inches; thus, A=lw. It's given that the area of a rectangle is 630 square inches and the length of the rectangle is 70 inches. Substituting 630 for A and 70 for l in the equation A=lw yields 630 = 70 w . Dividing both sides of this equation by 70 yields 9 = w . Therefore, the width, in inches, of this rectangle is 9 .

Choice B is incorrect. This is the length, not the width, in inches, of the rectangle.

Choice C is incorrect. This is half the area, in square inches, not the width, in inches, of the rectangle.

Choice D is incorrect. This is the difference between the area, in square inches, and the length, in inches, of the rectangle, not the width, in inches, of the rectangle.

Question 213 213 of 269 selected Area And Volume E

What is the area of a rectangle with a length of 17 centimeters (cm) and a width of 7 cm?

  1. 24  cm2

  2. 48  cm2

  3. 119  cm2

  4. 576  cm2

Show Answer Correct Answer: C

Choice C is correct. The area of a rectangle with length l  and width w can be found using the formula A=lw. It’s given that the rectangle has a length of 17 cm and a width of 7 cm. Therefore, the area of this rectangle is A=17(7), or 119 cm2.

Choice A is incorrect. This is the sum of the length and width of the rectangle, not the area.

Choice B is incorrect. This is the perimeter of the rectangle, not the area.

Choice D is incorrect. This is the sum of the length and width of the rectangle squared, not the area.

Question 214 214 of 269 selected Right Triangles And Trigonometry M

  • One angle is a right angle. 
  • The measure of a second angle is x°. 
  • The lengths of the 2 sides adjacent to the right angle are as follows: 
    • 32
    • 32
  • Note: Figure not drawn to scale. 

In the triangle shown, what is the value of x ?

Show Answer Correct Answer: 45

The correct answer is 45. An isosceles right triangle has a right angle and two legs of equal length. In the triangle shown, one angle is a right angle and the two legs each have a length of 32. Thus, the given triangle is an isosceles right triangle. In an isosceles right triangle, the measures of the two non-right angles are 45°. It follows that the value of x is 45.

Question 215 215 of 269 selected Right Triangles And Trigonometry M

The length of a rectangle’s diagonal is 5 17 , and the length of the rectangle’s shorter side is 5 . What is the length of the rectangle’s longer side?

  1. 17

  2. 20

  3. 15 2

  4. 400

Show Answer Correct Answer: B

Choice B is correct. A rectangle’s diagonal divides a rectangle into two congruent right triangles, where the diagonal is the hypotenuse of both triangles. It’s given that the length of the diagonal is 517 and the length of the rectangle’s shorter side is 5 . Therefore, each of the two right triangles formed by the rectangle’s diagonal has a hypotenuse with length 517, and a shorter leg with length 5 . To calculate the length of the longer leg of each right triangle, the Pythagorean theorem, a2+b2=c2, can be used, where a and b are the lengths of the legs and c is the length of the hypotenuse of the triangle. Substituting 5 for a and 517 for c in the equation a2+b2=c2  yields 52+b2=(517)2, which is equivalent to 25+b2=25(17), or 25+b2=425. Subtracting 25 from each side of this equation yields b2=400. Taking the positive square root of each side of this equation yields b=20. Therefore, the length of the longer leg of each right triangle formed by the diagonal of the rectangle is 20 . It follows that the length of the rectangle’s longer side is 20 .

Choice A is incorrect and may result from dividing the length of the rectangle’s diagonal by the length of the rectangle’s shorter side, rather than substituting these values into the Pythagorean theorem.

Choice C is incorrect and may result from using the length of the rectangle’s diagonal as the length of a leg of the right triangle, rather than the length of the hypotenuse.

Choice D is incorrect. This is the square of the length of the rectangle’s longer side.

Question 216 216 of 269 selected Area And Volume E

The area of a rectangle is 57 square inches. The length of the longest side of the rectangle is 19 inches. What is the length, in inches, of the shortest side of this rectangle?

Show Answer Correct Answer: 3

The correct answer is 3 . The area of a rectangle can be calculated by multiplying the length of its longest side by the length of its shortest side. It’s given that the area of the rectangle is 57 square inches and the length of the longest side of the rectangle is 19 inches. Let x represent the length, in inches, of the shortest side of this rectangle. It follows that 57=19x. Dividing both sides of this equation by 19 yields 3=x. Therefore, the length, in inches, of the shortest side of the rectangle is 3 .

Question 217 217 of 269 selected Circles H

Points A and B lie on a circle with radius 1, and arc A, B has length pi over 3. What fraction of the circumference of the circle is the length of arc A, B ?

Show Answer

The correct answer is one sixth. The circumference, C, of a circle is C equals, 2 pi, r, where r is the length of the radius of the circle. For the given circle with a radius of 1, the circumference is C equals, 2 pi, times 1, or C equals, 2 pi. To find what fraction of the circumference the length of arc A, B is, divide the length of the arc by the circumference, which gives the fraction pi over 3, end fraction, divided by 2 pi. This division can be represented by the fraction pi over 3, end fraction, times the fraction 1 over 2 pi, end fraction, equals one sixth. Note that 1/6, .1666, .1667, 0.166, and 0.167 are examples of ways to enter a correct answer.

Question 218 218 of 269 selected Right Triangles And Trigonometry H

A square is inscribed in a circle. The radius of the circle is 2022 inches. What is the side length, in inches, of the square?

  1. 20

  2. 2022

  3. 202

  4. 40

Show Answer Correct Answer: A

Choice A is correct. When a square is inscribed in a circle, a diagonal of the square is a diameter of the circle. It's given that a square is inscribed in a circle and the length of a radius of the circle is 2022 inches. Therefore, the length of a diameter of the circle is 2(2022) inches, or 202 inches. It follows that the length of a diagonal of the square is 202 inches. A diagonal of a square separates the square into two right triangles in which the legs are the sides of the square and the hypotenuse is a diagonal. Since a square has 4 congruent sides, each of these two right triangles has congruent legs and a hypotenuse of length 202 inches. Since each of these two right triangles has congruent legs, they are both 45 -45 -90 triangles. In a 45 -45 -90 triangle, the length of the hypotenuse is 2 times the length of a leg. Let s represent the length of a leg of one of these 45 -45 -90 triangles. It follows that 202=2(s). Dividing both sides of this equation by 2 yields 20=s. Therefore, the length of a leg of one of these 45 -45 -90 triangles is 20 inches. Since the legs of these two 45 -45 -90 triangles are the sides of the square, it follows that the side length of the square is 20 inches.

Choice B is incorrect. This is the length of a radius, in inches, of the circle.

Choice C is incorrect. This is the length of a diameter, in inches, of the circle.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 219 219 of 269 selected Right Triangles And Trigonometry H

Triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles C and F are right angles. The length of AB¯ is 2.9 times the length of DE¯. If tanA=2120, what is the value of sinD?

Show Answer Correct Answer: .7241, 21/29

The correct answer is 21 29 . It's given that triangle A B C is similar to triangle D E F , where angle A corresponds to angle D and angles C and F are right angles. In similar triangles, the tangents of corresponding angles are equal. Therefore, if tanA=2120, then tanD=2120. In a right triangle, the tangent of an acute angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Therefore, in triangle D E F , if tanD=2120, the ratio of the length of EF¯ to the length of DF¯ is 21 20 . If the lengths of EF¯ and DF¯ are 21 and 20 , respectively, then the ratio of the length of EF¯ to the length of DF¯ is 21 20 . In a right triangle, the sine of an acute angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Therefore, the value of sinD is the ratio of the length of EF¯ to the length of DE¯. The length of DE¯ can be calculated using the Pythagorean theorem, which states that if the lengths of the legs of a right triangle are a and b and the length of the hypotenuse is c , then a 2 + b 2 = c 2 . Therefore, if the lengths of EF¯ and DF¯ are 21 and 20 , respectively, then (21)2+(20)2=(DE)2, or 841=(DE)2. Taking the positive square root of both sides of this equation yields 29=DE. Therefore, if the lengths of EF¯ and DF¯ are 21 and 20 , respectively, then the length of DE¯ is 29 and the ratio of the length of EF¯ to the length of DE¯ is  21 29 . Thus, if tanA=2120, the value of sinD is 21 29 . Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.

Question 220 220 of 269 selected Area And Volume H

  • Moving from left to right, the points have the following coordinates:
    • (negative 3 comma 4)
    • (4 comma negative 3)
    • (5 comma 3)

What is the area, in square units, of the triangle formed by connecting the three points shown?

Show Answer Correct Answer: 24.5, 49/2

The correct answer is 24.5. It's given that a triangle is formed by connecting the three points shown, which are (-3,4)(5,3), and (4,-3). Let this triangle be triangle A. The area of triangle A can be found by calculating the area of the rectangle that circumscribes it and subtracting the areas of the three triangles that are inside the rectangle but outside triangle A. The rectangle formed by the points (-3,4)(5,4)(5,-3), and (-3,-3) circumscribes triangle A. The width, in units, of this rectangle can be found by calculating the distance between the points (5,4) and (5,-3). This distance is 4-(-3), or 7 . The length, in units, of this rectangle can be found by calculating the distance between the points (5,4) and (-3,4). This distance is 5-(-3), or 8 . It follows that the area, in square units, of the rectangle is (7)(8), or 56 . One of the triangles that lies inside the rectangle but outside triangle A is formed by the points (-3,4)(5,4), and (5,3). The length, in units, of a base of this triangle can be found by calculating the distance between the points (5,4) and (5,3). This distance is 4-3, or 1 . The corresponding height, in units, of this triangle can be found by calculating the distance between the points (5,4) and (-3,4). This distance is 5-(-3), or 8 . It follows that the area, in square units, of this triangle is 12(8)(1), or 4 . A second triangle that lies inside the rectangle but outside triangle A is formed by the points (4,-3), (5,3), and (5,-3). The length, in units, of a base of this triangle can be found by calculating the distance between the points (5,3) and (5,-3). This distance is 3-(-3) , or 6 . The corresponding height, in units, of this triangle can be found by calculating the distance between the points (5,-3) and (4,-3). This distance is 5-4, or 1 . It follows that the area, in square units, of this triangle is 12(1)(6), or 3 . The third triangle that lies inside the rectangle but outside triangle A is formed by the points (-3,4), (-3,-3), and (4,-3). The length, in units, of a base of this triangle can be found by calculating the distance between the points (4,-3) and (-3,-3). This distance is 4-(-3), or 7 . The corresponding height, in units, of this triangle can be found by calculating the distance between the points (-3,4) and (-3,-3). This distance is 4-(-3), or 7 . It follows that the area, in square units, of this triangle is 12(7)(7), or 24.5. Thus, the area, in square units, of the triangle formed by connecting the three points shown is 56-4-3-24.5, or 24.5. Note that 24.5 and 49/2 are examples of ways to enter a correct answer.

Question 221 221 of 269 selected Lines, Angles, And Triangles E

  • At the intersection of the 2 lines, the angles are labeled clockwise from top as follows:
    • Top: w°
    • Right: unlabeled
    • Bottom: z°
    • Left: unlabeled
  • A note indicates the figure is not drawn to scale.

 

In the figure, two lines intersect at a point. If w=136, what is the value of z ?

  1. 36

  2. 44

  3. 68

  4. 136

Show Answer Correct Answer: D

Choice D is correct. In the figure shown, the angles with measures w° and z° are vertical angles. Since vertical angles are congruent, w = z . Therefore, if w = 136 , the value of z is 136 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect. This is the measure, in degrees, of an angle that's supplementary, not congruent, to the angle with measure w°.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 222 222 of 269 selected Area And Volume H

Circle A  has a radius of 3 n and circle B  has a radius of 129 n , where n is a positive constant. The area of circle B  is how many times the area of circle A ?

  1. 43

  2. 86

  3. 129

  4. 1,849

Show Answer Correct Answer: D

Choice D is correct. The area of a circle can be found by using the formula A=πr2, where A is the area and r is the radius of the circle. It’s given that the radius of circle A is 3n. Substituting this value for r into the formula A=πr2 gives A=π(3n)2, or 9πn2. It’s also given that the radius of circle B is 129n. Substituting this value for r into the formula A=πr2 gives A=π(129n)2, or 16,641πn2. Dividing the area of circle B by the area of circle A gives 16,641πn29πn2, which simplifies to 1,849 . Therefore, the area of circle B is 1,849 times the area of circle A.

Choice A is incorrect. This is how many times greater the radius of circle B is than the radius of circle A.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the coefficient on the term that describes the radius of circle B.

Question 223 223 of 269 selected Area And Volume E

A triangle has a base length of 40 centimeters and a height of 90 centimeters. What is the area, in square centimeters, of the triangle?

Show Answer Correct Answer: 1800

The correct answer is 1,800 . The area, A , of a triangle can be found using the formula A=12bh, where b is the base length of the triangle and h is the height of the triangle. It’s given that the triangle has a base length of 40 centimeters and a height of 90 centimeters. Substituting 40 for b and 90 for h in the formula A=12bh yields A=12(40)(90), or A = 1,800 . Therefore, the area, in square centimeters, of the triangle is 1,800 .

Question 224 224 of 269 selected Right Triangles And Trigonometry H

In a right triangle, the tangent of one of the two acute angles is the fraction with numerator the square root of 3, and denominator 3. What is the tangent of the other acute angle?

  1. the negative of the fraction with numerator the square root of 3, and denominator 3

  2. the negative of the fraction with numerator 3, and denominator the square root of 3

  3. the fraction with numerator the square root of 3, and denominator 3

  4. the fraction with numerator 3, and denominator the square root of 3

Show Answer Correct Answer: D

Choice D is correct. The tangent of a nonright angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Using that definition for tangent, in a right triangle with legs that have lengths a and b, the tangent of one acute angle is the fraction a, over b and the tangent for the other acute angle is the fraction b over a. It follows that the tangents of the acute angles in a right triangle are reciprocals of each other. Therefore, the tangent of the other acute angle in the given triangle is the reciprocal of the fraction, the square root of 3, end root, over 3, end fraction or the fraction, 3 over the square root of 3, end fraction.

Choice A is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the tangent of the angle described. Choice B is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the reciprocal of the tangent of the angle described. Choice C is incorrect and may result from interpreting the tangent of the other acute angle as equal to the tangent of the angle described.

 

Question 225 225 of 269 selected Area And Volume M

A right circular cylinder has a base diameter of 22 centimeters and a height of 6 centimeters. What is the volume, in cubic centimeters, of the cylinder?

  1. 132π

  2. 264π

  3. 726π

  4. 2,904π

Show Answer Correct Answer: C

Choice C is correct. The volume, V , of a right circular cylinder is given by the formula V=πr2h, where r is the radius of the base of the cylinder and h is the height of the cylinder. It's given that a right circular cylinder has a height of 6 centimeters. Therefore, h = 6 . It's also given that the cylinder has a base diameter of 22 centimeters. The radius of a circle is half the diameter of the circle. Since the base of a right circular cylinder is a circle, it follows that the radius of the base of the right circular cylinder is 222, or 11 , centimeters. Therefore, r = 11 . Substituting 11 for r and 6 for h in the formula V=πr2h yields V=π(11)2(6), which is equivalent to V=π(121)(6), or V=726π. Therefore, the volume, in cubic centimeters, of the cylinder is 726π.

Choice A is incorrect. This is the volume of a right circular cylinder that has a base diameter of 222, not 22 , centimeters and a height of 6 centimeters.

Choice B is incorrect. This is the volume of a right circular cylinder that has a base diameter of 411, not 22 , centimeters and a height of 6 centimeters.

Choice D is incorrect. This is the volume of a right circular cylinder that has a base diameter of 44 , not 22 , centimeters and a height of 6 centimeters.

Question 226 226 of 269 selected Right Triangles And Trigonometry M
The figure presents triangle P Q R, where side P R is horizontal, and point Q is directly above point P. Angle P is a right angle. Angle R is labeled 30 degrees and side Q R is labeled 8.

In the right triangle shown above, what is the length of line P Q ?

Show Answer

The correct answer is 4. Triangle PQR has given angle measures of 30° and 90°, so the third angle must be 60° because the measures of the angles of a triangle sum to 180°. For any special right triangle with angles measuring 30°, 60°, and 90°, the length of the hypotenuse (the side opposite the right angle) is 2x, where x is the length of the side opposite the 30° angle. Segment PQ is opposite the 30° angle. Therefore, 2(PQ) = 8 and PQ = 4.

Question 227 227 of 269 selected Area And Volume E

Each side of a square has a length of 45 . What is the perimeter of this square?

Show Answer Correct Answer: 180

The correct answer is 180 . The perimeter of a polygon is equal to the sum of the lengths of the sides of the polygon. It’s given that each side of the square has a length of 45 . Since a square is a polygon with 4 sides, the perimeter of this square is 45+45+45+45, or 180 .

Question 228 228 of 269 selected Lines, Angles, And Triangles H

  • Counterclockwise from top right, the lines are labeled s, r, q, and t.
  • Line s intersects line r above line q.
  • Line s and line r each intersect line q and line t.
  • At the intersection of line s and line r, 1 angle is labeled counterclockwise from top as follows:
    • Top: a°
  • At the intersection of line s and line q, 1 angle is labeled counterclockwise from top-left as follows:
    • Top left: b°
  • At the intersection of line r and line t:
    • The line segment divides the bottom-left angle into two equal angles, each labeled w°.
  • A note indicates the figure is not drawn to scale.

In the figure, parallel lines q and t are intersected by lines r and s . If a = 43 and b = 122 , what is the value of w ?

Show Answer Correct Answer: 101/2, 50.5

The correct answer is 101 2 . In the figure, lines q , r , and s form a triangle. One interior angle of this triangle is vertical to the angle marked a°; therefore, the interior angle also has measure a°. It's given that a = 43 . Therefore, the interior angle of the triangle has measure 43°. A second interior angle of the triangle forms a straight line, q , with the angle marked b°. Therefore, the sum of the measures of these two angles is 180°. It's given that b = 122 . Therefore, the angle marked b° has measure 122° and the second interior angle of the triangle has measure (180-122)°, or 58°. The sum of the interior angles of a triangle is 180°. Therefore, the measure of the third interior angle of the triangle is (180-43-58)°, or 79°. It's given that parallel lines q and t are intersected by line r . It follows that the triangle's interior angle with measure 79° is congruent to the same side interior angle between lines q and t formed by lines t and r . Since this angle is supplementary to the two angles marked w°, the sum of 79°w°, and w° is 180°. It follows that 79+w+w=180, or 79+2w=180. Subtracting 79 from both sides of this equation yields 2w=101. Dividing both sides of this equation by 2 yields w = 101 2 . Note that 101/2 and 50.5 are examples of ways to enter a correct answer.

Question 229 229 of 269 selected Area And Volume E

What is the area of a rectangle with a length of 4 centimeters (cm) and a width of 2 cm?

  1. 6  cm2

  2. 8  cm2

  3. 12  cm2

  4. 36  cm2

Show Answer Correct Answer: B

Choice B is correct. The area of a rectangle with length l and width w can be found using the formula A=lw. It’s given that the rectangle has a length of 4 cm and a width of 2 cm. Therefore, the area of this rectangle is (4 cm)(2 cm), or 8 cm2.

Choice A is incorrect. This is the sum, in cm, of the length and width of the rectangle, not the area, in cm2.

Choice C is incorrect. This is the perimeter, in cm, of the rectangle, not the area, in cm2.

Choice D is incorrect. This is the sum of the length and width of the rectangle squared, not the area.

Question 230 230 of 269 selected Area And Volume H

  • The line segment slants gradually up from left to right.
  • The line segment begins at the point (negative 6 comma 4).
  • The line segment ends at the point (3 comma 10).

The line segment shown in the xy-plane represents one of the legs of a right triangle. The area of this triangle is 36 13 square units. What is the length, in units, of the other leg of this triangle?

  1. 12

  2. 24

  3. 3 13

  4. 18 13

Show Answer Correct Answer: B

Choice B is correct. The length of a segment in the xy-plane can be found using the distance formula, (x2-x1)2+(y2-y1)2, where (x1,y1) and (x2,y2) are the endpoints of the segment. The segment shown has endpoints at (-6,4) and (3,10). Substituting (-6,4) and (3,10) for (x1,y1) and (x2,y2), respectively, in the distance formula yields (3-(-6))2+(10-4)2, or 92+62, which is equivalent to 81+36, or 117. Let x represent the length, in units, of the other leg of this triangle. The area, A , of a right triangle can be calculated using the formula A=12bh, where b and h are the lengths of the legs of the triangle. It's given that the area of the triangle is 3613 square units. Substituting 3613 for A , 117 for b , and x for h in the formula A=12bh yields 3613=12(117)(x). Multiplying both sides of this equation by 2 yields 7213=x117. Dividing both sides of this equation by 117 yields 7213117=x. Multiplying the numerator and denominator on the left-hand side of this equation by 117 yields 721,521117=x, or 72(39)117=x, which is equivalent to 2,808117=x, or 24 = x . Therefore, the length, in units, of the other leg of this triangle is 24 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. 313 is equivalent to 117, which is the length, in units, of the line segment shown in the xy-plane, not the length, in units, of the other leg of the triangle.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 231 231 of 269 selected Right Triangles And Trigonometry H

  • Angle upper S is a right angle.
  • The length of side upper R upper T is 53.
  • A note indicates the figure is not drawn to scale.

In the triangle shown, RS=105. What is the value of sinR?

Show Answer Correct Answer: .9811, 52/53

The correct answer is 5253. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. The length of the hypotenuse of the right triangle shown is 53. It’s given that RS=105. Therefore, the length of one of the legs of the triangle shown is 105. Let x represent TS, the length of the other leg of the triangle shown. Therefore, 532=(105)2+x2, or 2,809=105+x2. Subtracting 105 from both sides of this equation yields 2,704=x2. Taking the positive square root of both sides of this equation yields 52=x. Therefore, TS, the length of the other leg of the triangle shown, is 52. The sine of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The length of the leg opposite angle R is 52, and the length of the hypotenuse is 53. Therefore, the value of sinR is 5253. Note that 52/53 or .9811 are examples of ways to enter a correct answer.

Question 232 232 of 269 selected Area And Volume E

A rectangle has a length of 64 inches and a width of 32 inches. What is the area, in square inches, of the rectangle?

Show Answer Correct Answer: 2048

The correct answer is 2,048 . The area A , in square inches, of a rectangle is equal to the product of its length l, in inches, and its width w , in inches, or A=lw. It's given that the rectangle has a length of 64 inches and a width of 32 inches. Substituting 64 for l and 32 for w in the equation A=lw yields A=(64)(32), or A = 2,048 . Therefore, the area, in square inches, of the rectangle is 2,048 .

Question 233 233 of 269 selected Lines, Angles, And Triangles H

  • Angle upper D upper E upper A is a right angle.
  • Point upper B lies on side upper A upper D.
  • Point upper C lies on side upper A upper E.
  • Line segment upper B upper C is drawn from side upper A upper D to side upper A upper E to form right triangle upper A upper B upper C.
  • Angle upper B upper C upper A is a right angle.
  • A note indicates the figure is not drawn to scale.

In the figure shown, AB=34 units, AC=3 units, and CE=21 units. What is the area, in square units, of triangle ADE?

Show Answer Correct Answer: 480

The correct answer is 480 . It's given in the figure that angle ACB and angle AED are right angles. It follows that angle ACB is congruent to angle AED. It's also given that angle BAC and angle DAE are the same angle. It follows that angle BAC is congruent to angle DAE. Since triangles A B C and A D E have two pairs of congruent angles, the triangles are similar. Sides A B and A C in triangle A B C correspond to sides A D and A E , respectively, in triangle A D E . Corresponding sides in similar triangles are proportional. Therefore, ADAB=AEAC. It's given that AC=3 units and CE=21 units. Therefore, AE=24 units. It’s also given that AB=34 units. Substituting 3 for A C , 24 for A E , and 34 for A B in the equation ADAB=AEAC yields AD34=243, or AD34=8. Multiplying each side of this equation by 34 yields AD=834. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b , then a2+b2=c2. Since triangle A D E is a right triangle, it follows that A D represents the length of the hypotenuse, c , and D E and A E represent the lengths of the legs, a and b . Substituting 24 for b and 834 for c in the equation a2+b2=c2 yields a2+(24)2=(834)2, which is equivalent to a2+576=64(34), or a2+576=2,176. Subtracting 576 from both sides of this equation yields a2=1,600. Taking the square root of both sides of this equation yields a=±40. Since a represents a length, which must be positive, the value of a is 40 . Therefore, D E = 40 . Since D E and A E represent the lengths of the legs of triangle A D E , it follows that D E and A E can be used to calculate the area, in square units, of the triangle as 12(40)(24), or 480 . Therefore, the area, in square units, of triangle A D E is 480 .

Question 234 234 of 269 selected Area And Volume H

The dimensions of a right rectangular prism are 4 inches by 5 inches by 6 inches. What is the surface area, in square inches, of the prism?

  1. 30

  2. 74

  3. 120

  4. 148

Show Answer Correct Answer: D

Choice D is correct. The surface area is found by summing the area of each face. A right rectangular prism consists of three pairs of congruent rectangles, so the surface area is found by multiplying the areas of three adjacent rectangles by 2 and adding these products. For this prism, the surface area is equal to 2 times, open parenthesis, 4 times 5, close parenthesis, plus, 2 times, open parenthesis, 5 times 6, close parenthesis, plus, 2 times, open parenthesis, 4 times 6, close parenthesis, or 2 times 20, plus, 2 times 30, plus, 2 times 24, which is equal to 148.

Choice A is incorrect. This is the area of one of the faces of the prism. Choice B is incorrect and may result from adding the areas of three adjacent rectangles without multiplying by 2. Choice C is incorrect. This is the volume, in cubic inches, of the prism.

Question 235 235 of 269 selected Right Triangles And Trigonometry H

Triangle A B C is similar to triangle D E F , where angle A corresponds to angle D and angle C corresponds to angle F .  Angles C and F are right angles. If tan ( A ) = 50 7 , what is the value of  tan ( E ) ?

Show Answer Correct Answer: .14, 7/50

The correct answer is 7 50 . It's given that triangle A B C is similar to triangle D E F , where angle A corresponds to angle D and angle C corresponds to angle F . In similar triangles, the tangents of corresponding angles are equal. Since angle A and angle D are corresponding angles, if tan(A)=507, then tan(D)=507. It's also given that angles C and F are right angles. It follows that triangle D E F is a right triangle with acute angles D and E . The tangent of one acute angle in a right triangle is the inverse of the tangent of the other acute angle in the triangle. Therefore, tan(E)=1tan(D). Substituting 50 7 for tan(D) in this equation yields tan(E)=1507, or tan(E)=750. Thus, if tan(A)=507, the value of tan(E) is 7 50 . Note that 7/50 and .14 are examples of ways to enter a correct answer.

Question 236 236 of 269 selected Lines, Angles, And Triangles E
The figure presents 5 lines labeled j, k, l, m, and n. Lines m and n are horizontal and line m lies above line n. Line j is slanted downward and to the right, and line k is slanted upward and to the right. Lines j, k, and top horizontal line m intersect, forming six angles. The angle below line m and to the left of line k and the angle below line m and to the right of line j are both labeled, a, degrees. Line k intersects the bottom horizontal line n, forming four angles. The angle above line n and to the left of line k is labeled 130 degrees. Line l is vertical and intersects lines m and n on the right side of the figure. Line j, line l, and the bottom horizontal line n intersect, forming six angles. The angle, above line n, to the left of vertical line l, and to the right of line j is labeled b degrees. There is a right angle symbol at an angle formed by horizontal line m and vertical line l.

In the figure above, lines m and n are parallel. What is the value of b ?

  1. 40

  2. 50

  3. 65

  4. 80

Show Answer Correct Answer: A

Choice A is correct. Given that lines m and n are parallel, the angle marked 130° must be supplementary to the leftmost angle marked a° because they are same-side interior angles. Therefore, 130° + = 180°, which yields a = 50°. Lines l and m intersect at a right angle, so lines j, l, and m form a right triangle where the two acute angles are a° and b°. The acute angles of a right triangle are complementary, so + = 90°, which yields 50° + = 90°, and b = 40.

Choice B is incorrect. This is the value of a, not b. Choice C is incorrect and may be the result of dividing 130° by 2. Choice D is incorrect and may be the result of multiplying b by 2.

Question 237 237 of 269 selected Circles H

  • The center of the circle is the point (negative 2 comma 0).
  • Clockwise from top, the circle passes through the following points:
    • (negative 2 comma 3)
    • (1 comma 0)
    • (negative 2 comma negative 3)
    • (negative 5 comma 0)

Circle A (shown) is defined by the equation (x+2)2+y2=9. Circle B (not shown) is the result of shifting circle A down 6 units and increasing the radius so that the radius of circle B is 2 times the radius of circle A. Which equation defines circle B?

  1. (x+2)2+(y+6)2=(4)(9)

  2. 2(x+2)2+2(y+6)2=9

  3. (x+2)2+(y-6)2=(4)(9)

  4. 2(x+2)2+2(y-6)2=9

Show Answer Correct Answer: A

Choice A is correct. According to the graph, the center of circle A has coordinates (-2,0), and the radius of circle A is 3 . It’s given that circle B is the result of shifting circle A down 6 units and increasing the radius so that the radius of circle B is 2 times the radius of circle A. It follows that the center of circle B is 6 units below the center of circle A. The point that's 6 units below (-2,0) has the same x-coordinate as (-2,0) and has a y-coordinate that is 6 less than the y-coordinate of (-2,0). Therefore, the coordinates of the center of circle B are (-2,0-6), or (-2,-6). Since the radius of circle B is 2 times the radius of circle A, the radius of circle B is (2)(3). A circle in the xy-plane can be defined by an equation of the form (x-h)2+(y-k)2=r2, where the coordinates of the center of the circle are (h,k) and the radius of the circle is r . Substituting -2 for h , -6 for k , and (2)(3) for r in this equation yields (x-(-2))2+(y-(-6))2=((2)(3))2, which is equivalent to (x+2)2+(y+6)2=(2)2(3)2, or (x+2)2+(y+6)2=(4)(9). Therefore, the equation (x+2)2+(y+6)2=(4)(9) defines circle B.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This equation defines a circle that’s the result of shifting circle A up, not down, by 6 units and increasing the radius.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 238 238 of 269 selected Lines, Angles, And Triangles E

In triangle A B C , AB=4,680 millimeters (mm) and BC=4,680 mm. Which statement is sufficient to prove that triangle A B C is equilateral?

  1. AC=4,680 mm

  2. AC=468 mm

  3. AC=46.8 mm

  4. AC=4.68 mm

Show Answer Correct Answer: A

Choice A is correct. In an equilateral triangle, all three sides have the same length. It’s given that in triangle A B C , AB=4,680 mm and BC=4,680 mm. Therefore, if AC=4,680 mm, then all three sides of triangle A B C have the same length, so triangle A B C is equilateral. Therefore, AC=4,680 mm is sufficient to prove that triangle A B C is equilateral.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 239 239 of 269 selected Right Triangles And Trigonometry H
The figure presents triangle A, C E, with horizontal side A, C. Side A, E, is perpendicular to side A, C, such that point E, is above point A. Point B, lies on horizontal side A, C, point D, lies on hypotenuse C E, and line segment B D, is drawn such that it is parallel to side A, E, and forms triangle C D B. Side A, E, is labeled 18, side B C, is labeled 8, side B D, is labeled 6, and a right angle symbol is at point A

In the figure above, line segment B D is parallel to line segment A, E. What is the length of line segment C E ?

Show Answer

The correct answer is 30. In the figure given, since side B D is parallel to side A, E and both segments are intersected by side C E, then angle BDC and angle AEC are corresponding angles and therefore congruent. Angle BCD and angle ACE are also congruent because they are the same angle. Triangle BCD and triangle ACE are similar because if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since triangle BCD and triangle ACE are similar, their corresponding sides are proportional. So in triangle BCD and triangle ACE, side B D corresponds to side A, E and side C D corresponds to side C E. Therefore, the length of side B D over the length of side C D, equals the length of side A, E over the length of side C E. Since triangle BCD is a right triangle, the Pythagorean theorem can be used to give the value of CD: 6 squared, plus 8 squared, equals the length of side C D squared. Taking the square root of each side gives the length of side C D equals 10. Substituting the values in the proportion the length of side B D over the length of side C D, equals the length of side A, E over the length of side C E yields 6 over 10, equals 18 over the length of C E. Multiplying each side by CE, and then multiplying by 10 over 6 yields the length of side C E equal to 30. Therefore, the length of side C E is 30.

Question 240 240 of 269 selected Area And Volume H

A hemisphere is half of a sphere. If a hemisphere has a radius of 27 inches, which of the following is closest to the volume, in cubic inches, of this hemisphere?

  1. 1,500

  2. 6,100

  3. 30,900

  4. 41,200

Show Answer Correct Answer: D

Choice D is correct. The volume, V, of a sphere is given by V=43πr3, where r is the radius of the sphere. Since a hemisphere is half of a sphere, it follows that the volume, V, of a hemisphere is given by V=(12)(43)πr3, or V=23πr3. Substituting 27 for r in this formula yields V=23π(27)3, which gives V=13,122π, or V is approximately equal to 41,223.98. Therefore, the choice that is closest to the volume, in cubic inches, of this hemisphere is 41,200.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 241 241 of 269 selected Circles H

A circle in the xy-plane has equation open parenthesis, x plus 3, close parenthesis, squared, plus, open parenthesis, y minus 1, close parenthesis, squared, equals 25. Which of the following points does NOT lie in the interior of the circle?

  1. negative 7 comma 3

  2. negative 3 comma 1

  3. zero comma zero

  4. 3 comma 2

Show Answer Correct Answer: D

Choice D is correct. The circle with equation open parenthesis, x plus 3, close parenthesis, squared, plus, open parenthesis, y minus 1, close parenthesis, squared, equals 25 has center with coordinates negative 3 comma 1 and radius 5. For a point to be inside of the circle, the distance from that point to the center must be less than the radius, 5. The distance between the point with coordinates 3 comma 2 and the point with coordinates negative 3 comma 1 is the square root of, open parenthesis, negative 3 minus 3, close parenthesis, squared, plus, open parenthesis, 1 minus 2, close parenthesis, squared, end root, equals, the square root of, open parenthesis, negative 6, close parenthesis, squared, plus, open parenthesis, negative 1, close parenthesis, squared, end root, which equals the square root of 37, which is greater than 5. Therefore, the point with coordinates 3 comma 2 does NOT lie in the interior of the circle.

Choice A is incorrect. The distance between the point with coordinates negative 7 comma 3 and the point with coordinates negative 3 comma 1 is the square root of, open parenthesis, negative 7 plus 3, close parenthesis, squared, plus, open parenthesis, 3 minus 1, close parenthesis, squared, end root, equals, the square root of, open parenthesis, negative 4, close parenthesis, squared, plus, open parenthesis, 2, close parenthesis, squared, end root, which equals the square root of 20, which is less than 5, and therefore the point with coordinates negative 7 comma 3 lies in the interior of the circle. Choice B is incorrect because it is the center of the circle. Choice C is incorrect because the distance between the point with coordinates 0 comma 0 and the point with coordinates negative 3 comma 1 is the square root of, open parenthesis, 0 plus 3, close parenthesis, squared, plus, open parenthesis, 0 minus 1, close parenthesis, squared, end root, equals, the square root of, open parenthesis, 3, close parenthesis, squared, plus open parenthesis, 1, close parenthesis, squared, end root, which equals the square root of 8, which is less than 5, and therefore the point with coordinates 0 comma 0 in the interior of the circle.

 

Question 242 242 of 269 selected Lines, Angles, And Triangles M

Right triangles L M N and P Q R are similar, where L and M correspond to P and Q , respectively. Angle M has a measure of 53 °. What is the measure of angle Q ?

  1. 37 °

  2. 53 °

  3. 127 °

  4. 143 °

Show Answer Correct Answer: B

Choice B is correct. It’s given that triangle L M N is similar to triangle P Q R . Corresponding angles of similar triangles are congruent. Since angle M and angle Q correspond to each other, they must be congruent. Therefore, if the measure of angle M is 53°, then the measure of angle Q is also 53°.

Choice A is incorrect and may result from concluding that angle M and angle Q are complementary rather than congruent.

Choice C is incorrect and may result from concluding that angle M and angle Q are supplementary rather than congruent.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 243 243 of 269 selected Right Triangles And Trigonometry H

In triangle J K L , cos ( K ) =2451 and angle J is a right angle. What is the value of cos ( L ) ?

Show Answer Correct Answer: .8823, .8824, 15/17

The correct answer is 1517. It's given that angle J is the right angle in triangle JKL. Therefore, the acute angles of triangle JKL are angle K and angle L . The hypotenuse of a right triangle is the side opposite its right angle. Therefore, the hypotenuse of triangle JKL is side KL. The cosine of an acute angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. It's given that cos(K)=2451. This can be written as cos(K)=817. Since the cosine of angle K is a ratio, it follows that the length of the side adjacent to angle K is 8 n and the length of the hypotenuse is 17 n , where n is a constant. Therefore, JK=8n and KL=17n. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For triangle JKL, it follows that (JK)2+(JL)2=(KL)2. Substituting 8 n for J K and 17 n for K L yields (8n)2+(JL)2=(17n)2. This is equivalent to 64n2+(JL)2=289n2. Subtracting 64n2 from each side of this equation yields (JL)2=225n2. Taking the square root of each side of this equation yields JL=15n. Since cos(L)=JLKL, it follows that cos(L)=15n17n, which can be rewritten as cos(L)=1517. Note that 15/17, .8824, .8823, and 0.882 are examples of ways to enter a correct answer.

Question 244 244 of 269 selected Lines, Angles, And Triangles E

  • The 2 triangles are oriented such that upper A corresponds to upper X, upper B corresponds to upper Y, and upper C corresponds to upper Z.
  • Triangle upper A upper B upper C is labeled as follows:
    • The measure of angle upper A is 60°.
    • The length of side upper A upper C is d.
  • A note indicates the figures are not drawn to scale.

For the triangles shown, triangle A B C is dilated by a scale factor of 3 to obtain triangle X Y Z , where d=16. What is the measure, in degrees, of angle X ?

  1. 20

  2. 57

  3. 60

  4. 63

Show Answer Correct Answer: C

Choice C is correct. It's given that triangle XYZ is obtained by a dilation of triangle ABC. It follows that triangle ABC is similar to triangle XYZ, where A corresponds to X. Since corresponding angles in similar triangles have the same measure and the measure of angle A is 60 degrees, it follows that the measure of angle X is also 60 degrees.

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.  

Question 245 245 of 269 selected Area And Volume M

A right circular cylinder has a volume of 432 cubic centimeters. The area of the base of the cylinder is 24 square centimeters. What is the height, in centimeters, of the cylinder?

  1. 18

  2. 24

  3. 216

  4. 10,368

Show Answer Correct Answer: A

Choice A is correct. The volume, V , of a right circular cylinder is given by the formula V=πr2h, where πr2 is the area of the base of the cylinder and h is the height. It’s given that a right circular cylinder has a volume of 432 cubic centimeters and the area of the base is 24 square centimeters. Substituting 432 for V and 24 for πr2 in the formula V=πr2h yields 432 = 24 h . Dividing both sides of this equation by 24 yields 18 = h . Therefore, the height of the cylinder, in centimeters, is 18 .

Choice B is incorrect. This is the area of the base, in square centimeters, not the height, in centimeters, of the cylinder.

Choice C is incorrect. This is the height, in centimeters, of a cylinder if its volume is 432 cubic centimeters and the area of its base is 2 , not 24 , cubic centimeters.

Choice D is incorrect. This is the height, in centimeters, of a cylinder if its volume is 432 cubic centimeters and the area of its base is 124, not 24 , cubic centimeters.

Question 246 246 of 269 selected Lines, Angles, And Triangles E

In a right triangle, the measure of one of the acute angles is 51°. What is the measure, in degrees, of the other acute angle?

  1. 6

  2. 39

  3. 49

  4. 51

Show Answer Correct Answer: B

Choice B is correct. The sum of the measures of the interior angles of a triangle is 180 degrees. Since the triangle is a right triangle, it has one angle that measures 90 degrees. Therefore, the sum of the measures, in degrees, of the remaining two angles is 180-90, or 90 . It’s given that the measure of one of the acute angles in the triangle is 51 degrees. Therefore, the measure, in degrees, of the other acute angle is 90-51, or 39 .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the measure, in degrees, of the acute angle whose measure is given.

Question 247 247 of 269 selected Area And Volume E
The figure presents a right rectangular prism. The length of the prism is 6 centimeters, the width of the prism is 2 centimeters, and the height of the prism is 3 centimeters.

The figure shows the lengths, in centimeters (cm), of the edges of a right rectangular prism. The volume V of a right rectangular prism is l w h, where l is the length of the prism, w is the width of the prism, and h is the height of the prism. What is the volume, in cubic centimeters, of the prism?

  1. 36

  2. 24

  3. 12

  4. 11

Show Answer Correct Answer: A

Choice A is correct. It’s given that the volume of a right rectangular prism is l w h. The prism shown has a length of 6 cm, a width of 2 cm, and a height of 3 cm. Thus, l w h equals, 6 times 2, times 3, or 36 cubic centimeters.

Choice B is incorrect. This is the volume of a rectangular prism with edge lengths of 6, 2, and 2. Choice C is incorrect and may result from only finding the product of the length and width of the base of the prism. Choice D is incorrect and may result from finding the sum, not the product, of the edge lengths of the prism.

Question 248 248 of 269 selected Right Triangles And Trigonometry H

  • One angle is a right angle.
  • The measure of a second angle is x°.
  • The length of the hypotenuse is 28.
  • The length of the leg adjacent to the angle with measure x° is 11.
  • A note indicates the figure is not drawn to scale.

In the triangle shown, what is the value of cosx°?

Show Answer Correct Answer: .3928, .3929, 11/28

The correct answer is 1128. The cosine of an acute angle in a right triangle is defined as the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. In the triangle shown, the length of the leg adjacent to the angle with measure x° is 11 units and the length of the hypotenuse is 28 units. Therefore, the value of cosx° is 1128. Note that 11/28, .3928, .3929, 0.392, and 0.393 are examples of ways to enter a correct answer.

Question 249 249 of 269 selected Right Triangles And Trigonometry H

For two acute angles, Q and R, cos(Q)=sin(R). The measures, in degrees, of Q and R are x + 61 and 4 x + 4 , respectively. What is the value of x ?

  1. 5

  2. 19

  3. 23

  4. 29

Show Answer Correct Answer: A

Choice A is correct. It's given that for two acute angles, Q and R, cos(Q)=sin(R). For two acute angles, if the sine of one angle is equal to the cosine of the other angle, the angles are complementary. It follows that Q and R are complementary. That is, the sum of the measures of the angles is 90 degrees. It's given that the measure of Q is x+61 degrees and the measure of R is 4x+4 degrees. It follows that (x+61)+(4x+4)=90. By combining like terms, this equation can be rewritten as 5x+65=90. Subtracting 65 from each side of this equation yields 5x=25. Dividing each side of this equation by 5 yields x=5.

Choice B is incorrect. This would be the value of x if cos(Q)=cos(R) rather than cos(Q)=sin(R).

Choice C is incorrect. This would be the value of x if cos(Q)=-cos(R) rather than cos(Q)=sin(R) and if R were obtuse rather than acute.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 250 250 of 269 selected Area And Volume M

A cylindrical can containing pieces of fruit is filled to the top with syrup before being sealed. The base of the can has an area of 75 centimeters squared, and the height of the can is 10 cm. If 110 centimeters cubed of syrup is needed to fill the can to the top, which of the following is closest to the total volume of the pieces of fruit in the can?

  1. 7 point 5 centimeters cubed

  2. 185 centimeters cubed

  3. 640 centimeters cubed
  4. 750 centimeters cubed
Show Answer Correct Answer: C

Choice C is correct. The total volume of the cylindrical can is found by multiplying the area of the base of the can, 75 square centimeters, by the height of the can, 10 cm, which yields 750 cubic centimeters. If the syrup needed to fill the can has a volume of 110 cubic centimeters, then the remaining volume for the pieces of
fruit is 750 minus 110, equals 640 cubic centimeters.

Choice A is incorrect because if the fruit had a volume of 7 point 5 cubic centimeters, there would be 750 minus 7 point 5, equals 742 point 5 cubic centimeters of syrup needed to fill the can to the top. Choice B is incorrect because if the fruit had a volume of 185 cubic centimeters, there would be 750 minus 185, equals 565 cubic centimeters of syrup needed to fill the can to the top. Choice D is incorrect because it is the total volume of the can, not just of the pieces of fruit.

 

Question 251 251 of 269 selected Area And Volume M

A triangle has a base length of 56 centimeters and a height of 112 centimeters. What is the area, in square centimeters, of the triangle?

  1. 168

  2. 1,568

  3. 3,136

  4. 6,272

Show Answer Correct Answer: C

Choice C is correct. The area, A, of a triangle is given by the formula A=12bh, where b is the base length and h is the height of the triangle. It’s given that a triangle has a base length of 56 centimeters and a height of 112 centimeters. Substituting 56 for b and 112 for h in the formula A=12bh yields A=(12)(56)(112), or A=3,136. Therefore, the area, in square centimeters, of the triangle is 3,136.

Choice A is incorrect. This is the value of 56+112, not (12)(56)(112).

Choice B is incorrect. This is the value of (14)(56)(112), not (12)(56)(112).

Choice D is incorrect. This is the value of (56)(112), not (12)(56)(112).

Question 252 252 of 269 selected Right Triangles And Trigonometry E

  • One angle is a right angle.
  • The measure of the other 2 angles are as follows:
    • 45°
    • 45°
  • The lengths of the 2 sides adjacent to the right angle are as follows:
    • x
    • 24
  • Note: Figure not drawn to scale.

In the triangle shown, what is the value of x ?

  1. 24

  2. 45

  3. 48

  4. 69

Show Answer Correct Answer: A

Choice A is correct. Since the two acute angles have the same measure and the third angle is a right angle, the triangle shown is an isosceles right triangle. In an isosceles right triangle, the two legs have the same length. The figure shows that the length of one leg of the triangle is 24 and the length of the other leg of the triangle is x. It follows that the value of x is 24.

Choice B is incorrect. This is the measure, in degrees, of one of the angles shown.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Question 253 253 of 269 selected Lines, Angles, And Triangles E

The figure presents a triangle with a horizontal base. The left and right sides of the triangle are both extended past the top vertex of the triangle. The bottom left angle is labeled 70 degrees, and the bottom right angle is labeled 50 degrees. The angle above the left side of the triangle and below the extension of the right side of the triangle is labeled x degrees.

In the figure above, two sides of a triangle are extended. What is the value of x ?

  1. 110

  2. 120

  3. 130

  4. 140

Show Answer Correct Answer: B

Choice B is correct. The sum of the interior angles of a triangle is 180°. The measures of the two interior angles of the given triangle are shown. Therefore, the measure of the third interior angle is 180° – 70° – 50° = 60°. The angles of measures x° and 60° are supplementary, so their sum is 180°. Therefore, x = 180 – 60 = 120.

Choice A is incorrect and may be the result of misinterpreting x° as supplementary to 70°. Choice C is incorrect and may be the result of misinterpreting x° as supplementary to 50°. Choice D is incorrect and may be the result of a calculation error.

Question 254 254 of 269 selected Area And Volume E

The width of a rectangle is 7 centimeters. The length of the rectangle is 40 centimeters longer than the width. What is the area, in square centimeters, of this rectangle?

  1. 7

  2. 14

  3. 54

  4. 329

Show Answer Correct Answer: D

Choice D is correct. It’s given that the width of this rectangle is 7 centimeters and that the length of this rectangle is 40 centimeters longer than the width. Therefore, the length of this rectangle is 7+40, or 47, centimeters. The area of a rectangle can be found by multiplying its length and its width. Therefore the area, in square centimeters, of this rectangle is (7)(47), or 329.

Choice A is incorrect. This is the width, in centimeters, not the area, in square centimeters, of this rectangle.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Question 255 255 of 269 selected Lines, Angles, And Triangles M

In triangle A B C , the measure of angle A is 54°, the measure of angle B is 90°, and the measure of angle C is (k2)°. What is the value of k ?

  1. 36

  2. 45

  3. 72

  4. 108

Show Answer Correct Answer: C

Choice C is correct. The sum of the interior angles of a triangle is 180°. It's given that the interior angles of triangle A B C are 54°, 90°, and (k2)°. It follows that 54+90+k2=180, or 144+k2=180. Subtracting 144 from each side of this equation yields k2=36. Multiplying each side of this equation by 2 yields k = 72 . Therefore, the value of k is 72 .

Choice A is incorrect. This is the value of k 2 , not k .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 256 256 of 269 selected Area And Volume H

A right triangle has sides of length 2 2 , 6 2 , and 80 units. What is the area of the triangle, in square units?

  1. 8 2 +80

  2. 12

  3. 24 80

  4. 24

Show Answer Correct Answer: B

Choice B is correct. The area, A , of a triangle can be found using the formula A=12bh, where b is the length of the base of the triangle and h is the height of the triangle. It's given that the triangle is a right triangle. Therefore, its base and height can be represented by the two legs. It’s also given that the triangle has sides of length 22, 62, and 80 units. Since 80 units is the greatest of these lengths, it's the length of the hypotenuse. Therefore, the two legs have lengths 22 and 62 units. Substituting these values for b and h in the formula A=12bh gives A=12(22)(62), which is equivalent to A=64 square units, or A = 12 square units.

Choice A is incorrect. This expression represents the perimeter, rather than the area, of the triangle.

Choice C is incorrect and may result from conceptual or calculation errors. 

Choice D is incorrect and may result from conceptual or calculation errors. 

Question 257 257 of 269 selected Lines, Angles, And Triangles H
The figure presents right triangle A B C . Side A C is horizontal, vertex C is to the right of vertex A, vertex B is above side A C , and angle A B C is a right angle. Point D lies on side A C, directly below vertex B, and vertical line segment B D is drawn, dividing triangle A B C into two smaller triangles, triangle A B D and triangle B D C. Angle A D B is a right angle. A note states that the figure is not drawn to scale.

In the figure above, the length of line segment B D equals 6 and the length of line segment A D equals 8. What is the length of line segment D C ?

Show Answer

The correct answer is 4.5. According to the properties of right triangles, BD divides triangle ABC into two similar triangles, ABD and BCD. The corresponding sides of ABD and BCD are proportional, so the ratio of BD to AD is the same as the ratio of DC to BD. Expressing this information as a proportion gives six eighths equals, the fraction D C, over 6. Solving the proportion for DC results in D C equals 4 point 5. Note that 4.5 and 9/2 are examples of ways to enter a correct answer.

Question 258 258 of 269 selected Area And Volume M

The figure presents a cylindrical shape with a circular base and a larger circular top. The diameter of the circular base is labeled “k over 2,” the diameter of the circular top is labeled “k,” and the height is labeled “k.” The volume of the figure is equal to the fraction with numerator 7 pi k cubed, and denominator 48

The glass pictured above can hold a maximum volume of 473 cubic centimeters, which is approximately 16 fluid ounces. What is the value of k, in centimeters?

 

  1.   2.52

  2.   7.67

  3.   7.79

  4. 10.11

Show Answer Correct Answer: D

Choice D is correct. Using the volume formula V equals, the fraction with numerator 7 pi times k cubed, and denominator 48 and the given information that the volume of the glass is 473 cubic centimeters, the value of k can be found as follows:

473 equals, the fraction with numerator 7 pi times k cubed, and denominator 48

k cubed equals, the fraction with numerator 473 times 48, and denominator 7 pi, end fraction

k equals, the cube root of the fraction with numerator 473 times 48, and denominator 7 pi, end fraction, end root, which is approximately equal to 10 point 1 0 6 9 0

Therefore, the value of k is approximately 10.11 centimeters.

Choices A, B, and C are incorrect. Substituting the values of k from these choices in the formula results in volumes of approximately 7 cubic centimeters, 207 cubic centimeters, and 217 cubic centimeters, respectively, all of which contradict the given information that the volume of the glass is 473 cubic centimeters.

 

Question 259 259 of 269 selected Lines, Angles, And Triangles H

A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of the acute angles is (9x-560)°. The sum of the measures of one of the acute angles and three of the obtuse angles is (-18x+w)°. What is the value of w ?

Show Answer Correct Answer: 1660

The correct answer is 1,660 . It’s given that a line intersects two parallel lines, forming four acute angles and four obtuse angles. When two parallel lines are intersected by a transversal line, the angles formed have the following properties: two adjacent angles are supplementary, and alternate interior angles are congruent. Therefore, each of the four acute angles have the same measure, and each of the four obtuse angles have the same measure. It’s also given that the measure of one of the acute angles is (9x-560)°. If two angles are supplementary, then the sum of their measures is 180°. Therefore, the measure of the obtuse angle adjacent to any of the acute angles is (180-(9x-560))°, or (180-9x+560)°, which is equivalent to (-9x+740)°. It’s given that the sum of the measures of one of the acute angles and three of the obtuse angles is (18x+w)°. It follows that (9x560)+3(9x+740)=(18x+w), which is equivalent to 9x56027x+2,220=18x+w, or -18x+1,660=-18x+w. Adding 18 x to both sides of this equation yields 1,660=w

Question 260 260 of 269 selected Area And Volume E

What is the perimeter, in inches, of a rectangle with a length of 4 inches and a width of 9 inches?

  1. 13

  2. 17

  3. 22

  4. 26

Show Answer Correct Answer: D

Choice D is correct. The perimeter of a figure is equal to the sum of the measurements of the sides of the figure. It’s given that the rectangle has a length of 4 inches and a width of 9 inches. Since a rectangle has 4 sides, of which opposite sides are parallel and equal, it follows that the rectangle has two sides with a length of 4 inches and two sides with a width of 9 inches. Therefore, the perimeter of this rectangle is 4+4+9+9, or 26 inches.

Choice A is incorrect. This is the sum, in inches, of the length and the width of the rectangle.

Choice B is incorrect. This is the sum, in inches, of the two lengths and the width of the rectangle.

Choice C is incorrect. This is the sum, in inches, of the length and the two widths of the rectangle.

Question 261 261 of 269 selected Circles M

A circle in the xy-plane has its center at (-4,5) and the point (-8,8) lies on the circle. Which equation represents this circle?

  1. (x-4)2+(y+5)2=5

  2. (x+4)2+(y-5)2=5

  3. (x-4)2+(y+5)2=25

  4. (x+4)2+(y-5)2=25

Show Answer Correct Answer: D

Choice D is correct. A circle in the xy-plane can be represented by an equation of the form (x-h)2+(y-k)2=r2, where (h,k) is the center of the circle and r is the length of a radius of the circle. It's given that the circle has its center at (-4,5). Therefore, h = -4 and k = 5 . Substituting -4 for h and 5 for k in the equation (x-h)2+(y-k)2=r2 yields (x-(-4))2+(y-5)2=r2, or (x+4)2+(y-5)2=r2. It's also given that the point (-8,8) lies on the circle. Substituting -8 for x and 8 for y in the equation (x+4)2+(y-5)2=r2 yields (-8+4)2+(8-5)2=r2, or (-4)2+(3)2=r2, which is equivalent to 16+9=r2, or 25=r2. Substituting 25 for r 2 in the equation (x+4)2+(y-5)2=r2 yields (x+4)2+(y-5)2=25. Thus, the equation (x+4)2+(y-5)2=25 represents the circle.

Choice A is incorrect. The circle represented by this equation has its center at (4,-5), not (-4,5), and the point (-8,8) doesn't lie on the circle.

Choice B is incorrect. The point (-8,8) doesn't lie on the circle represented by this equation.

Choice C is incorrect. The circle represented by this equation has its center at (4,-5), not (-4,5), and the point (-8,8) doesn't lie on the circle.

Question 262 262 of 269 selected Area And Volume H

Triangles ABC and DEF are similar. Each side length of triangle ABC is 4 times the corresponding side length of triangle DEF. The area of triangle ABC is 270 square inches. What is the area, in square inches, of triangle DEF?

Show Answer Correct Answer: 135/8, 16.87, 16.88

The correct answer is 135 8 . It's given that triangles ABC and DEF are similar and each side length of triangle ABC is 4 times the corresponding side length of triangle DEF. For two similar triangles, if each side length of the first triangle is k times the corresponding side length of the second triangle, then the area of the first triangle is k 2 times the area of the second triangle. Therefore, the area of triangle ABC is 42, or 16 , times the area of triangle DEF. It's given that the area of triangle ABC is 270 square inches. Let a represent the area, in square inches, of triangle DEF. It follows that 270 is 16 times a , or 270 = 16 a . Dividing both sides of this equation by 16 yields 27016=a, which is equivalent to 1358=a. Thus, the area, in square inches, of triangle DEF is 135 8 . Note that 135/8, 16.87, and 16.88 are examples of ways to enter a correct answer.

Question 263 263 of 269 selected Circles H

x 2 + 14 x + y 2 = 6 y + 109

In the xy-plane, the graph of the given equation is a circle. What is the length of the circle's radius?

  1. 109

  2. 149

  3. 167

  4. 341

Show Answer Correct Answer: C

Choice C is correct. It's given that in the xy-plane, the graph of the given equation is a circle. The equation of a circle in the xy-plane can be written in the form (x-h)2+(y-k)2=r2, where (h,k) is the center of the circle and r is the length of the circle's radius. Subtracting 6 y from both sides of the equation x2+14x+y2=6y+109 yields x2+14x+y2-6y=109. By completing the square, this equation can be rewritten as (x2+14x+(142)2)+(y2-6y+(-62)2)=109+(142)2+(-62)2. This equation can be rewritten as (x2+14x+49)+(y2-6y+9)=109+49+9, or (x+7)2+(y-3)2=167. Therefore, r 2 = 167 . Taking the square root of both sides of this equation yields r=167 and r=-167. Since r is the length of the circle's radius, r must be positive. Therefore, the length of the circle's radius is 167.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Question 264 264 of 269 selected Lines, Angles, And Triangles E

  • Clockwise from top left, the 3 lines are labeled t, p, and q.
  • Line t intersects both line p and line q.
  • At the intersection of line t and line q, 2 angles are labeled clockwise from top left as follows:
    • Top left: x°
    • Top right: 142°
  • A note indicates the figure is not drawn to scale.

In the figure, line p is parallel to line q , and line t intersects both lines. What is the value of x+142?

  1. 52

  2. 90

  3. 142

  4. 180

Show Answer Correct Answer: D

Choice D is correct. In the figure shown, the angle marked x° and the angle marked 142° form a linear pair of angles. If two angles form a linear pair of angles, the sum of the measures of the angles is 180°. Therefore, the value of x + 142 is 180 .

Choice A is incorrect. This is 90 less than 142 , not the sum of x and 142 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the measure, in degrees, of one of the angles shown.

Question 265 265 of 269 selected Circles H

What is the value of sin42π?

  1. 0

  2. 1 2

  3. 2 2

  4. 1

Show Answer Correct Answer: A

Choice A is correct. The sine of a number t is the y-coordinate of the point arrived at by traveling a distance of t units counterclockwise around the unit circle from the starting point (1,0). Since the unit circle has a circumference of 2π units, it follows that one full rotation around the circle is equal to a distance of 2π units. Therefore, a distance of 42π units around the circle from the starting point (1,0) would result in exactly 21 full rotations, arriving back at the point (1,0). So, sin42π is equal to the y-coordinate of the point (1,0), which is 0 .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the value of cos42π, not sin42π.

Question 266 266 of 269 selected Right Triangles And Trigonometry H

A triangle with angle measures 30°, 60°, and 90° has a perimeter of 18 plus 6, times, the square root of 3. What is the length of the longest side of the triangle?

Show Answer

The correct answer is 12. It is given that the triangle has angle measures of 30°, 60°, and 90°, and so the triangle is a special right triangle. The side measures of this type of special triangle are in the ratio 2 to 1 to the square root of 3. If x is the measure of the shortest leg, then the measure of the other leg is the square root of 3 end root x and the measure of the hypotenuse is 2x. The perimeter of the triangle is given to be 18, plus 6 times, the square root of 3, end root, and so the equation for the perimeter can be written as 2 x, plus x, plus the square root of 3 end root x, equals 18, plus 6 times, the square root of 3. Combining like terms and factoring out a common factor of x on the left-hand side of the equation gives open parenthesis, 3 plus the square root of 3, close parenthesis, times x, equals 18, plus 6 times the square root of 3. Rewriting the right-hand side of the equation by factoring out 6 gives open parenthesis, 3 plus the square root of 3, close parenthesis, times x, equals 6 times, open parenthesis, 3 plus the square root of 3, close parenthesis. Dividing both sides of the equation by the common factor open parenthesis, 3 plus the square root of 3, close parenthesis gives x = 6. The longest side of the right triangle, the hypotenuse, has a length of 2x, or 2(6), which is 12.

Question 267 267 of 269 selected Area And Volume H

A cube has a surface area of 54 square meters. What is the volume, in cubic meters, of the cube?

  1. 18

  2. 27

  3. 36

  4. 81

Show Answer Correct Answer: B

Choice B is correct. The surface area of a cube with side length s is equal to 6 s squared. Since the surface area is given as 54 square meters, the equation 54 equals, 6 s squared can be used to solve for s. Dividing both sides of the equation by 6 yields 9 equals, s squared. Taking the square root of both sides of this equation yields 3 equals s and negative 3, equals s. Since the side length of a cube must be a positive value, s equals, negative 3 can be discarded as a possible solution, leaving s equals 3. The volume of a cube with side length s is equal to s cubed. Therefore, the volume of this cube, in cubic meters, is 3 cubed, or 27.

Choices A, C, and D are incorrect and may result from calculation errors.

 

Question 268 268 of 269 selected Area And Volume M
The figure presents the graph of square A B C D in the x y plane. The numbers negative 5 and 5 are indicated on each axis. The coordinates of each vertex are as follows.
Vertex A has coordinates 0 comma negative 5.
Vertex B has coordinates negative 5 comma 0.
Vertex C has coordinates 0 comma 5.
Vertex D has coordinates 5 comma 0.

In the xy-plane shown, square ABCD has its diagonals on the x- and y-axes. What is the area, in square units, of the square?

  1. 20

  2. 25

  3. 50

  4. 100

Show Answer Correct Answer: C

Choice C is correct. The two diagonals of square ABCD divide the square into 4 congruent right triangles, where each triangle has a vertex at the origin of the graph shown. The formula for the area of a triangle is A equals, one half times b h, where b is the base length of the triangle and h is the height of the triangle. Each of the 4 congruent right triangles has a height of 5 units and a base length of 5 units. Therefore, the area of each triangle is A equals, one half times, 5 times 5, or 12.5 square units. Since the 4 right triangles are congruent, the area of each is one fourth of the area of square ABCD. It follows that the area of the square ABCD is equal to 4 times 12 point 5, or 50 square units.

Choices A and D are incorrect and may result from using 5 or 25, respectively, as the area of one of the 4 congruent right triangles formed by diagonals of square ABCD. However, the area of these triangles is 12.5. Choice B is incorrect and may result from using 5 as the length of one side of square ABCD. However, the length of a side of square ABCD is 5 times the square root of 2.

Question 269 269 of 269 selected Circles M

In the xy-plane, the graph of the equation ( x - 3 ) 2 + ( y - 5 ) 2 = 9 is a circle. The point (6,c), where c is a constant, lies on this circle. What is the value of c ?

Show Answer Correct Answer: 5

The correct answer is 5. It's given that in the xy-plane, the graph of the equation (x-3)2+(y-5)2=9 is a circle. It’s also given that the point (6,c), where c is a constant, lies on this circle. It follows that the ordered pair (6,c) makes the equation (x-3)2+(y-5)2=9 true. Substituting 6 for x and c for y in this equation yields (6-3)2+(c-5)2=9, or 9+(c-5)2=9. Subtracting 9 from each side of this equation yields (c-5)2=0. It follows that the value of c is 5.